case study of co ordinate geometry

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CBSE Class 10 Maths Case Study Questions for Chapter 7 - Coordinate Geometry (Released By CBSE)

Cbse's question bank on case study for class 10 maths chapter 7 is available here. practice this new type of questions to score good marks in your board exam..

Case study based questions for CBSE Class 10 Maths Chapter 7 - Coordinate Geometry are provided here for students to practice this new format of questions for their Maths Board Exam 2022. All these questions are published by the Central Board of Secondary Education (CBSE) for Class 10 Maths. Students must solve these questions to familiarise themselves with the concepts and logic used in the case study. You can also check the right answer at the end of each question.

Check Case Study Questions for Class 10 Maths Chapter 7 - Coordinate Geometry

CASE STUDY 1:

In order to conduct Sports Day activities in your School, lines have been drawn with chalk powder at a distance of 1 m each, in a rectangular shaped ground ABCD, 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in given figure below. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th distance AD on the eighth line and posts a red flag.

1. Find the position of green flag

b) (2,0.25)

d) (0, -25)

Answer:  a) (2,25)

2. Find the position of red flag

Answer: c) (8,20)

3. What is the distance between both the flags?

a) √41

b) √11

c) √61

d) √51

Answer: c) √61

4. If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

a) (5, 22.5)

d) (2.5,20)

Answer: a) (5, 22.5)

5. If Joy has to post a flag at one-fourth distance from green flag , in the line segment joining the green and red flags, then where should he post his flag?

a) (3.5,24)

b) (0.5,12.5)

c) (2.25,8.5)

Answer: a) (3.5,24)

CASE STUDY 2:

The class X students school in krishnagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

1. Taking A as origin, find the coordinates of P

Answer: a) (4,6)

2. What will be the coordinates of R, if C is the origin?

Answer: c) (10,3)

3. What will be the coordinates of Q, if C is the origin?

b) b) (-6,13)

Answer: d) (13,6)

4. Calculate the area of the triangles if A is the origin

Answer: a) 4.5

5. Calculate the area of the triangles if C is the origin

Answer: d) 4.5

Also Check:

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Class 9th Maths - Coordinate Geometry Case Study Questions and Answers 2022 - 2023

By QB365 on 08 Sep, 2022

QB365 provides a detailed and simple solution for every Possible Case Study Questions in Class 9th Maths Subject - Coordinate Geometry, CBSE. It will help Students to get more practice questions, Students can Practice these question papers in addition to score best marks.

QB365 - Question Bank Software

Coordinate geometry case study questions with answer key.

9th Standard CBSE

Final Semester - June 2015

Mathematics

(b) What are the coordinates of C and D respectively?

(c) What is the distance between B and D?

(d) What is the distance between A and C?

(e) What are the coordinates of the point of intersection of AC and BD?

(ii) What are the coordinates of Police Station?

(iii) Distance between school and police station:

(iv) What are the coordinates of Library?

(v) In which quadrant the point (-1, 4) lies?  

(b) What are the coordinates of A and B respectively?

(c) The coordinates of point O in the sketch -2 is

(d) The point on the y-axis ( in sketch 2) which is equidistant from the points B and C is 

(e) The point on the x-axis ( in sketch 2) which is equidistant from the points C and D is

(b) What are the coordinates of R, taking A as origin?

(c) Side of lawn is :

(d) Shape of lawn is :

(e) Area of lawn is :

(ii) What are the coordinates of position 'D'?

(iii) What are the coordinates of position 'H'?

(iv) In which quadrant, the point 'C' lie?

(v) Find the perpendicular distance of the point E from the y-axis.

*****************************************

Coordinate geometry case study questions with answer key answer keys.

(a) (iii) A(3, 5); B(7, 9) (b) (i) C(11, 5); D(7, 1) (c) (iii) 8 units (d) (iii) 8 units (e) (i) (7, 5)

(i) (b) (2, 3) (ii) (a) (2, -1) (iii) (a) 4 (iv) (d) (6, 2) (v) (b) II

(a) (ii) A(13, 10); B(19, 10) (b) (iv) A(19, 6); B(13, 6) (c) (ii) (16, 8) (d) (i) (0, 8) (e)  (ii) (16, 0)

(a) (iv) C(10, 6) (b) (iii) R(5, 6) (c) (ii)   \(\sqrt{34}\)  units PS 2 = AS 2 + AP 2 = 5 2 + 3 2 = 25 + 9 = 34 ⇒ PS =  \(\sqrt{34}\) (d) (iv) Rhombus (e) (i) 30 sq. units Area of rhombus =  \(1 / 2\)  x product of diagonals =  \(1 / 2\)  x 6 x 10  = 30 sq. units

(i) (d) (-4, 3)  (ii) (b) (-3, -2) (iii) (b) (8, 4.5) (iv) (d) IV (v) (b) 10 units

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Class 10 Maths Case Study Questions Chapter 7 Coordinate Geometry

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Case study Questions in the Class 10 Mathematics Chapter 7  are very important to solve for your exam. Class 10 Maths Chapter 7 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving  Class 10 Maths Case Study Questions  Chapter 7  Coordinate Geometry

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

In CBSE Class 10 Maths Paper, Students will have to answer some questions based on Assertion and Reason. There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Coordinate Geometry Case Study Questions With Answers

Here, we have provided case-based/passage-based questions for Class 10 Maths  Chapter 7 Coordinate Geometry

Case Study/Passage-Based Questions

Question 1:

Answer: (d) none of these

(ii) The distance of the bus stand from the house is

Answer: (b) 10 cm

(iii) If the grocery store and electrician’s shop lie on a line, the ratio of the distance of the house from the grocery store to that from the electrician’s shop, is

Answer: (c) 1.2

(iv) The ratio of distances of the house from the bus stand to food cart is

Answer: (c) 1.1

(v) The coordinates of positions of bus stand, grocery store, food cart, and electrician’s shop form a

Question 2:

The class X student’s school in krishnagar has been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

1. Taking A as origin, find the coordinates of P

Answer: a) (4,6)

2. What will be the coordinates of R, if C is the origin?

Answer: c) (10,3)

3. What will be the coordinates of Q, if C is the origin?

b) b) (-6,13)

Answer: d) (13,6)

4. Calculate the area of the triangles if A is the origin

Answer: a) 4.5

5. Calculate the area of the triangles if C is the origin

Answer: d) 4.5

Hope the information shed above regarding Case Study and Passage Based Questions for Class 10 Maths Chapter 7 Coordinate Geometry with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 10 Maths Coordinate Geometry Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate

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Class 10 Maths Chapter 7 Case Based Questions - Coordinate Geometry

Study Case - 1

The class X students of a school in Rajinder Nagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Mango are planted on the boundary at the distance of 1 m from each other.

Based on the above figure, answer the following questions:

  Q4: What will be the coordinates of the vertices of ΔPQR if C is the origin? (a) (14, 3), (11, 2), (8, 6) (b) (15, 4), (12, 3), (9, 7) (c) (14, 2), (11, 1), (8, 5) (d) (15, 3), (12, 2), (9, 6) Ans:  (d) Explanation:  When C is taken as origin, we will take CB as X-axis and CD as Y-axis. Then, coordinates of points, P, Q and R are (15, 3), (12, 2)and (9, 6) respectively.

Q5: What are the coordinates of P if D is taken as the origin? (a) (−15, 5) (b) (15, 5) (c) (15, 7) (d) (15, 3) Ans:  (a) Explanation:  When D is taken as origin, we will take DA as negative X-axis and DC as positive Y-axis.   Then coordinates of point P is (−15, 5).

Case Study - 2

Based on the above information, give the answer of the following questions: 

Study Case - 3

Based on the above information give the answer of the following questions:

Q1: The coordinates of the point A are: (a) (4,9) (b) (5,9) (c) (92,9) (d) (4,8) Ans: (c) Explanation:  As the distance of point A is 9/2 units from the Y-axis and 9 units from the X-axis, its coordinates are x = 9/2, y = 9 or (9/2, 9)

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Case Study Questions for Class 9 Maths Chapter 3 Coordinate Geometry

• Last modified on: 1 year ago
• Reading Time: 2 Minutes

Case Study Questions

Question 1:

Saumya has to reach her office every day at 10:00 am. On the way to her office, she drops her son at school. Now, the location of Saumya’s house, her son’s school and her office are represented by the map below. Using the details given, answer the following questions.

Q1. Find the coordinates of Saumya’s home. (a) (1, 4) (b) (4, 1) (c) (7, 1) (d) (1, 7)

Q2. Find the coordinates of Saumya’s office. (a) (7, 5) (b) (5, 7) (c) (7, 1) (d) (1, 7)

Q3. Find the coordinates of Saumya’s son’s school. (a) (1, 4) (b) (4, 1) (c) (7, 1) (d) (1, 7)

Q4. Find the distance between Saumya’s home and her son’s school. (a) 7km (b) 4km (c) 3km (d) 1km

Q5. Find the distance between Saumya’s office and her son’s school. (a) 7km (b) 4km (c) 3km (d) 1km

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• Class 10 Maths
• Chapter 7: Coordinate Geometry

Important Questions Class 10 Maths Chapter 7 Coordinate Geometry

Important Questions Class 10 Maths Chapter 7 Coordinate Geometry are given at BYJU’S with stepwise solutions. Students who are preparing for the board exams of 2022-23 are advised to practice these important questions of Coordinate Geometry to score high marks in the Maths exam. These questions will provide good practice to students to solve any problem asked in the exam from this chapter. Solving these questions will also help in revision, and it will boost the confidence level of students.

This chapter has some important formulas to help students with their board exam preparation and competitive exams. Some of the important questions from the distance formula, section formula, and the triangle area are provided here. Students can also get the solutions for all the questions of the Class 10 NCERT textbook for Maths.

• Coordinate Geometry Formulas
• Coordinate Geometry For Class 10
• Coordinate System

Important Questions & Answers For Class 10 Maths Chapter 7 Coordinate Geometry

Q. 1: Find the distance of the point P (2, 3) from the x-axis.

We know that,

(x, y) = (2, 3) is a point on the Cartesian plane in the first quadrant.

x = Perpendicular distance from y-axis

y = Perpendicular distance from x-axis

Therefore, the perpendicular distance from x-axis = y coordinate = 3

Q. 2: Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

Let P(x, y) be equidistant from the points A(7, 1) and B(3, 5).

Then, AP = BP

AP 2 = BP 2

Using distance formula,

(x – 7) 2 + (y – 1) 2 = (x – 3) 2 + (y – 5) 2

x 2 – 14x + 49 + y 2 – 2y + 1 = x 2 – 6x + 9 + y 2 – 10y + 25

Hence, the relation between x and y is x – y = 2.

Q. 3: Find the coordinates of the points of trisection (i.e., points dividing into three equal parts) of the line segment joining the points A(2, – 2) and B(– 7, 4).

Let P and Q be the points of trisection of AB, i.e., AP = PQ = QB.

Therefore, P divides AB internally in the ratio 1: 2.

Let (x 1 , y 1 ) = (2, -2)

(x 2 , y 2 ) = (-7, 4)

m 1 : m 2 = 1 : 2

Therefore, the coordinates of P, by applying the section formula,

= (-3/3, 0/3)

Similarly, Q also divides AB internally in the ratio 2 : 1. and the coordinates of Q by applying the section formula,

= (-12/3, 6/3)

Hence, the coordinates of the points of trisection of the line segment joining A and B are (–1, 0) and (– 4, 2).

Q. 4: Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).

Let the ratio in which the line segment joining ( -3, 10) and (6, -8) is divided by point ( -1, 6) be k:1.

Therefore by section formula,

-1 = ( 6k-3)/(k+1)

–k – 1 = 6k -3

Hence, the required ratio is 2 : 7.

Q. 5: Find the value of k if the points A(2, 3), B(4, k) and C(6, –3) are collinear.

A(2, 3)= (x 1 , y 1 )

B(4, k) = (x 2 , y 2 )

C(6, -3) = (x 3 , y 3 )

If the given points are collinear, the area of the triangle formed by them will be 0.

½ [x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )] = 0

½ [2(k + 3) + 4(-3 -3) + 6(3 – k)] = 0

½ [2k + 6 – 24 + 18 – 6k] = 0

½ (-4k) = 0

Q. 6: Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

Let the vertices of the triangle be A(0, -1), B(2, 1) and C(0, 3).

Let D, E, F be the mid-points of the sides of this triangle.

Using the mid-point formula, coordinates of D, E, and F are:

D = [(0+2)/2, (-1+1)/2] = (1, 0)

E = [(0+0)/2, (-1+3)/2] = (0, 1)

F = [(0+2)/2, (3+1)/2] = (1, 2)

Area of triangle DEF = ½ {(1(2 – 1) + 1(1 – 0) + 0(0 – 2)}

= ½ (1 + 1)

Area of triangle DEF = 1 sq.unit

Area of triangle ABC = ½ {0(1 – 3) + 2(3 – (-1)) + 0(-1 – 1)}

Area of triangle ABC = 4 sq.units

Hence, the ratio of the area of triangle DEF and ABC = 1 : 4.

Q. 7: Name the type of triangle formed by the points A (–5, 6), B (–4, –2) and C (7, 5).

The points are A (–5, 6), B (–4, –2) and C (7, 5).

d = √ ((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 )

AB = √((-4+5)² + (-2-6)²)

BC=√((7+4)² + (5+2)²)

=√(121 + 49)

AC=√((7+5)² + (5-6)²)

Since all sides are of different lengths, ABC is a scalene triangle.

Q.8: Find the area of triangle PQR formed by the points P(-5, 7), Q(-4, -5) and R(4, 5).

P(-5, 7), Q(-4, -5) and R(4, 5)

Let P(-5, 7) = (x 1 , y 1 )

Q(-4, -5) = (x 2 , y 2 )

R(4, 5) = (x 3 , y 3 )

Area of the triangle PQR = (½)|x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )|

= (½) |-5(-5 – 5) + (-4)(5 – 7) + 4(7 + 5)|

= (½) |-5(-10) -4(-2) + 4(12)|

= (½) |50 + 8 + 48|

= (½) × 106

Therefore, the area of triangle PQR is 53 sq. units.

Q.9: If the point C(-1, 2) divides internally the line segment joining A(2, 5) and B(x, y) in the ratio 3 : 4, find the coordinates of B.

C(-1, 2) divides internally the line segment joining A(2, 5) and B(x, y) in the ratio 3 : 4.

A(2, 5) = (x 1 , y 1 ) 

B(x, y) = (x 2 , y 2 )

m : n = 3 : 4

Using section formula,

C(-1, 2) = [(mx 2 + nx 1 )/(m + n), (my 2 + ny 1 )/(m + n)]

= [(3x + 8)/(3 + 4), (3y + 20)/(3 + 7)]

By equating the corresponding coordinates,

(3x + 8)/7 = -1

3x + 8 = -7

3x = -7 – 8

(3y + 20)/7 = 2

3y + 20 = 14

3y = 14 – 20

Therefore, the coordinates of B(x, y) = (-5, -2).

Q.10: Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (-2, -5) and (6, 3). Find the coordinates of the point of intersection.

Let the given points be:

A(-2, -5) = (x 1 , y 1 )

B(6, 3) = (x 2 , y 2 )

The line x – 3y = 0 divides the line segment joining the points A and B in the ratio k : 1.

Point of division P(x, y) = [(kx 2 + x 1 )/(k + 1), (ky 2 + y 1 )/(k + 1)]

x = (6k – 2)/(k + 1) and y = (3k – 5)/(k + 1)

Here, the point of division lies on the line x – 3y = 0.

[(6k – 2)/(k + 1)] – 3[(3k – 5)/(k + 1)] = 0

6k – 2 – 3(3k – 5) = 0

6k – 2 – 9k + 15 = 0

-3k + 13 = 0

Thus, the ratio in which the line x – 3y = 0 divides the line segment AB is 13 : 3.

Therefore, x = [6(13/3) – 2]/ [(13/3) + 1]

= (78 – 6)/(13 + 3)

y = [3(13/3) – 5]/ [(13/3) + 1]

= (39 – 15)/(13 + 3)

Therefore, the coordinates of the point of intersection = (9/2, 3/2).

Q.11: Write the coordinates of a point on the x-axis which is equidistant from points A(-2, 0) and B(6, 0).

Let P(x, 0) be a point on the x-axis.

Given that point, P is equidistant from points A(-2, 0) and B(6, 0).

Squaring on both sides,

(AP)² = (BP)²

(x + 2)² + (0 – 0)² = (x – 6)² + (0 – 0)²

x² + 4x + 4 = x² – 12x + 36

4x + 12x = 36 – 4

Therefore, the coordinates of a point on the x-axis = (2, 0).

Q.12: If A(-2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence, find the lengths of its sides.

Given vertices of a parallelogram ABCD are:

A(-2, 1), B(a, 0), C(4, b) and D(1, 2)

We know that the diagonals of a parallelogram bisect each other.

So, midpoint of AC = midpoint of BD

[(-2 + 4)/2, (1 + b)/2] = [(a + 1)/2, (0 + 2)/2]

2/2 = (a + 1)/2 and (1 + b)/2 = 2/2

a + 1 = 2 and b + 1 = 2

a = 1 and b = 1

Therefore, a = 1 and b = 1.

Let us find the lengths of sides of a parallelogram, i.e. AB, BC, CD and DA

Using the distance formula,

AB = √[(1 + 2)² + (0 – 1)²] = √(9 + 1) = √10 units

BC = √[(4 – 1)² + (1 – 0)²] = √(9 + 1) = √10 units

And CD = √10 and DA = √10 {the opposite sides of a parallelogram are parallel and equal}

Hence, the length of each side of the parallelogram ABCD = √10 units.

Q.13: If A(-5, 7), B(-4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

Given vertices of a quadrilateral are:

A(-5, 7), B(-4, -5), C(-1, -6) and D(4, 5)

The quadrilateral ABCD can be divided into two triangles ABD and BCD.

Area of the triangle with vertices (x 1 , y 1 ), (x 2 , y 2 ), and (x 3 , y 3 ) = (½) |x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )|

Area of triangle ABD = (½) |-5(-5 – 5) + (-4)(5 – 7) + 4(7 + 5)|

Area of triangle BCD = (½) |-4(-6 – 5) + (-1)(5 + 5) + 4(-5 + 6)|

= (½) |-4(-11) -1(10) + 4(1)|

= (½) |44 – 10 + 4|

Therefore, the area of quadrilateral ABCD = Area of triangle ABD + Area of triangle BCD

= 72 sq.units

Q.14: Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3). Hence, find m.

Let P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3) in the ratio k : 1.

P(4, m) = (x, y)

A(2, 3) = (x 1 , y 1 )

B(6, -3) = (x 2 , y 2 )

p(x, y) = [(kx 2 + x 1 )/(k + 1), (ky 2 + y 1 )/(k + 1)]

(4, m) = [(6k + 2)/(k + 1), (-3k + 3)/(k + 1)]

By equating the x-coodinate,

(6k + 2)/(k + 1) = 4

6k + 2 = 4k + 4

6k – 4k = 4 – 2

Thus, the point P divides the line segment joining A and B in the ratio 1 : 1.

Now by equating the y-coodinate,

(-3k + 3)/(k + 1) = m

Substituting k = 1,

[-3(1) + 3]/(1 + 1) = m

m = (3 – 3)/2

Q.15: Find the distance of a point P(x, y) from the origin.

Coordinates of origin = O(0, 0)

Let P(x, y) = (x 1 , y 1 )

O(0, 0) = (x 2 , y 2 )

OP = √[(x 2 – x 1 )² + (y 2 – y 1 )²]

= √[(x – 0)² + (y – 0)²]

= √(x² + y²)

Hence, the distance of the point P(x, y) from the origin is √(x² + y²) units.

Practice Questions For Class 10 Maths Chapter 7 Coordinate Geometry

• The centre of a circle is (2a, a – 7). Find the values of a, if the circle passes through the point (11, –9) and has a diameter 10√ 2 units.
• Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1). Also, find the coordinates of the point of division.
• Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square.
• Find the point on the x-axis, which is equidistant from (2, –5) and (–2, 9).
• Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order. [ Hint : Area of a rhombus = 1/2 (product of its diagonals)]
• If the points A (1, –2), B (2, 3) C (a, 2) and D (– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as the base.
• Find a relation between x and y if the points A(x, y), B(-4, 6) and C(-2, 3) are collinear.
• Find the area of a triangle whose vertices are given as (1, -1), (-4, 6) and (-3, -5).
• If the distances of P(x, y) from A(5, 1) and B(-1, 5) are equal, then prove that 3x = 2y.
• In what ratio does the point (24/11, y) divide the line segment joining the points P(2, -2) and Q(3, 7)? Also, find the value of y.

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CBSE Case Study Questions for Class 9 Maths Coordinate Geometry Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 9 Maths Coordinate Geometry  in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

CBSE Case Study Questions for Class 9 Maths Coordinate Geometry PDF

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Case Study Chapter 7 Coordinate Geometry Mathematics

Refer to Case Study Chapter 7 Coordinate Geometry Mathematics, these class 10 maths case study based questions have been designed as per the latest examination guidelines issued for the current academic year by CBSE, NCERT, KVS. Students should go through these solves case studies so that they are able to understand the pattern of questions expected in exams and get good marks.

Chapter 7 Coordinate Geometry Mathematics Case Study Based Questions

l. Read the following and answer Shaurya made a map of his locality on a coordinate plane.   

Question. The coordinates of the point which divides the line segment joining school and park internally in the ratio 3 : 2 are    (a) (–2, 2) (b) (–2, –2) (c) (2, 3) (d) (2, –2)

Question. If he considered his house as the origin, then coordinates of market are    (a) (3, –1) (b) (–3, –1) (c) (–3, 1) (d) (3, 1)

Question. The distance of his friend Kartik’s house from his house is    (a) √20 units (b) √10 units (c) 20 units (d) 10 units

Question. If you form a polygon with vertex as position of park, Shaurya’s home, railway station, post office and temple, then the polygon is    (a) Regular polygon (b) Convex Polygon (c) Concave Polygon (d) Rhombus

Question. There is a fort at a distance of 10 units from his house. If its ordinate is 6, then its abscissa is    (a) ± 2 (b) 0 (c) ± 4 (d) ± 8

ll. Read the following and answer A coach is discussing the strategy of the game with his players. The position of players is marked with ‘×’ in the figure.   

Question. The player who is 6 units from x-axis and 2 units to the right of y-axis is at position      (a) J (b) B (c) I (d) A

Question. According to sudden requirement coach of the team decided to increase one player in the 4th quadrant without increasing the total number of players, so he decided to change the position of player F in such a way that F becomes symmetric to D w.r.t x axis, then new position of F is    (a) (3, 4) (b) (3, – 4) (c) (–4, 3) (d) (4, 3)

Question. If (x, y) are the coordinates of the mid-point of the line segment joining A and H, then      (a) x = – 4, y = 2 (b) x = 2, y = 4 (c) x = –2, y = 4 (d) x = –4, y = – 2

Question. If O is taken as the origin, the point whose abscissa equal to zero is      (a) H (b) E (c) G (d) F

Question. The distance between the player C and B is      (a) 5 units  (b) 4 √2 units (c) 2 √5 units (d) 5 √2 units

lll. Read the following and answer The children of a school prepared a dance item for Republic Day parade for which they were asked to form a rectangle by standing at a fixed distance, taken as one unit. Some children, then formed a pattern inside the rectangle. 

Question. The coordinate of the point that divides the line segment joining the points A and D in the ratio 2 : 3 internally are      (a) (6, 5 /19)  (b) (6, 6) (c) (6, 2) (d) (19/5 , 6)

Question. The coordinates of the point P if H is taken as the origin are      (a) (2, 3) (b) (–1, –3) (c) (–2, 3) (d) (2, –3)

Question. If a point (x, y) is equidistant from C(6, 8) and F(6, 1) then      (a) 2x – 7y + 36 = 0 (b) 144y = 63 (c) x – y = 5 (d) x + y = 5

Question. If P is considered as the origin, the coordinates of B are    (a) (8, 5) (b) (3, 8) (c) (8, 0) (d) (0, 3)

Question. The distance between the children standing at H and G is    (a) 8 units   (b) 2 units (c) 5 units (d) √8 units

lV. Read the following and answer The top of a table is shown in the figure given below: 

Question. The coordinates of the mid point of line segment joining points M and Q are    (a) (9, 3) (b) (5, 11) (c) (14, 14) (d) (7, 7)

Question. If G is taken as the origin, and x, y axis put along GF and GB, then the point denoted by coordinate (4, 2) is    (a) H (b) F (c) Q (d) R

Question. Which among the following have same ordinate ?    (a) H and A (b) T and O (c) R and M (d) N and R

Question. The coordinates of the points H and G are respectively    (a) (1, 5), (5, 1) (b) (0, 5), (5, 0) (c) (1, 5), (5, 0) (d) (5, 1), (1, 5)

Question. The distance between the points A and B is      (a) 4 units (b) 4 √2 units (c) 16 units (d) 32 units

V. Read the following and answer A rangoli design was made by Ishita using coordinate plane. 

Question. PQRS is a square inside the circle where P is (–1, 1) then coordinates of R are      (a) (–1, –1) (b) (–1, 1) (c) (1, –1) (d) (1, 1)

Question. The distance of the point M on the circle from x-axis is      (a) 4 units (b) 3 units (c) 2 units (d) 5 units

Question. The coordinates of the mid point of the line segment joining PR is      (a) (1, 1) (b) (0, 0) (c) (–1, –1) (d) (1, 2)

Question. If coordinates of centre X are (0, 0) and B is a point on circle with coordinates (7, 0), then coordinate of C and D are respectively    (a) (0, 7), (0, – 7) (b) (0, –7), (0, 7) (c) (7, 7), (–7, – 7) (d) (–7, –7), (7, 7)

Question. The coordinates of the point on the circle in first quadrant whose abscissa equal to 3 is    (a) (3, 3)  (b) (3, –3) (c) (2√10, 3) (d) (3, 2 √10)

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Chapter 7 Class 10 Coordinate Geometry

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Updated for new NCERT Book for 2023-2024 Boards.

Get NCERT Solutions of all Exercise Questions and Examples of Chapter 7 Class 10 Coordinate Geometry. All questions have been solved in an easy to understand way with detailed explanation of each and every step.

Answers to optional exercise is also given

In Coordinate Geometry of Class 9, we learned what is x and y coordinate of a point. 

In this chapter, we will learn 

• Coordinates of points in x-axis (x, 0) and y-axis (0, y)
• How to find distance between two points using Distance Formula
• Checking if points form a triangle, or an isoceles triangle, or a square, rectangle. (To do this, we need to learn properties of parallelogram )
• Checking if 3 points are collinear using Distance Formula
• Finding points equidistant to given points
• Then, we will learn about Section Formula - Finding coordinates of a point that divides line joining two points in some ratio
• Finding coordinates of mid-point of a line using Section Formula
• Finding ratio when coordinates of point of intersection is given
• Using properties of parallelogram to do some questions on Section Formula (like finding coordinates of a point, when 4 vertices of parallelogram are given)
• Finding Area of triangle when coordinates of all 3 vertices are given
• Proving 3 points collinear using Area of Triangle Formula
• Finding Area of Quadrialteral using Area of Triangle Formula (dividing Quadrilateral into 2 triangles and finding area)

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CBSE Class 10 Maths: Case Study Questions of Chapter 7 Coordinate Geometry PDF Download

Case study Questions in the Class 10 Mathematics Chapter 7  are very important to solve for your exam. Class 10 Maths Chapter 7 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving case study-based   questions for Class 10 Maths Chapter 7  Coordinate Geometry

In CBSE Class 10 Maths Paper, Students will have to answer some questions based on  Assertion and Reason . There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Coordinate Geometry Case Study Questions With answers

Here, we have provided case-based/passage-based questions for Class 10 Maths  Chapter 7 Coordinate Geometry

Case Study/Passage-Based Questions

Question 1:

Answer: (d) none of these

(ii) The distance of the bus stand from the house is

Answer: (b) 10 cm

(iii) If the grocery store and electrician’s shop lie on a line, the ratio of the distance of the house from the grocery store to that from the electrician’s shop, is

Answer: (c) 1.2

(iv) The ratio of distances of the house from the bus stand to food cart is

Answer: (c) 1.1

(v) The coordinates of positions of bus stand, grocery store, food cart, and electrician’s shop form a

Question 2:

The class X student’s school in krishnagar has been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

1. Taking A as origin, find the coordinates of P

Answer: a) (4,6)

2. What will be the coordinates of R, if C is the origin?

Answer: c) (10,3)

3. What will be the coordinates of Q, if C is the origin?

b) b) (-6,13)

Answer: d) (13,6)

4. Calculate the area of the triangles if A is the origin

Answer: a) 4.5

5. Calculate the area of the triangles if C is the origin

Answer: d) 4.5

Hope the information shed above regarding Case Study and Passage Based Questions for Class 10 Maths Chapter 7 Coordinate Geometry with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 10 Maths Coordinate Geometry Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate

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Co-ordinate Geometry’s application in real life

The study of geometry using coordinate points is known as coordinate geometry (or analytic geometry). It is a portion of geometry in which the position of a point is determined by coordinates. A series of values called coordinates is used to show the exact location of a point in the coordinate plane.

It’s always better to understand how information benefits us in the real world. Let’s look at how the co-ordinate geometry can be applied in practice.

Coordinate geometry applications

A few examples of coordinate geometry applications are shown below.

• It’s used to figure out how far two points are apart.
• It is used to calculate the dividing line ratio in the m:n ratio.
• It is used to find the line’s midpoint.
• It’s used to figure out how big a triangle in the Cartesian plane is.
• It’s used in physics, GPS, maps, and a variety of other fields under various names. The underlying principle is the same in every situation: you choose an origin, assign a measure of unit distance, and define two or three directions, and bingo, you can now locate any point! After a couple more points are fixed, do a few extra amazing things.
• Even a computer monitor employs some basic coordinate geometry concepts. Complex curves, conics, and shapes can be mathematically specified using algebraic equations, which would be impossible to analyse using pure geometry.

Some of the detailed applications are shown below.

The Digital Use

The text file or PDF file is an example of coordinate plane usage. The words or graphics in these files are written or edited using coordinate geometry. A 2-dimensional coordinate (x, y) system is used to read any PDF file that comprises text and graphics. Coordinate geometry is also used by scanners and photocopying machines to generate a perfect copy of the original image.

The Global Positioning System (GPS) is a satellite navigation system that delivers position and timing information in all weather circumstances.

Nature, which surrounds mankind, is the most essential example of geometry in everyday life. Different geometrical shapes and patterns can be found in leaves, flowers, stems, roots, bark, and so on if one looks closely. Geometry is also important in the human digestive system since it is organised like a tube within a tube. Tree leaves come in a variety of forms, sizes, and symmetries. Different geometrical shapes are found in different fruits and vegetables; for example, an orange is a sphere, and after peeling it, one can see how the individual slices make a perfect spherical shape.

Determining Latitude and Longitude

A coordinate framework is required for defining the precise location of a place in the actual world. To assign geographic locations, a particular coordinate system known as the geographical coordinate system is utilised. When weather forecasters watch storms in real life, they note the absolute position on a regular basis to see the storm’s route and try to anticipate the storm’s future path based in part on these discoveries.

Locating Air Transport

Coordinate geometry is used to manage and regulate air traffic. The coordinates of the flight are used to describe the aircraft’s current location. Even if an aircraft moves a tiny distance (up, down, forward, or backward), the system updates the coordinates of flight for every small change in its position.

Defining the location of any object

A coordinate system can be used to determine the position of any item from its starting point (known as the origin) to its current location.

For example, we can measure the distance between the watch and the television from the other side of the room. Allow a horizontal gap of 10 metre between the television and the corner, and a vertical distance of 5 metre between the watch and the ground. If the room is 5 metres wide, we can easily locate the watch’s coordinates and use the distance calculation to calculate its real distance.

Projection of Maps

Map Any 3D curved object can be projected onto a flat 2D surface via projection. This approach can be used to portray the curved surface of the earth on a flat map, for example. A unique sort of coordinate system known as a projected coordinate system is utilised for these purposes. These are primarily used to display maps on a computer screen.

Technology is the most widespread use of geometry in everyday life. Geometry is used in almost all basic principles in robotics, computers, and video games. Computer programmers can work because they have access to geometric principles at all times. Geometric computations aid in the production of sophisticated graphics in video games, which is how the virtual world of video games is generated. Raycasting is a shooting technique that uses a 2-D map to stimulate the 3-D environment of video games. Raycasting aids in processing speed by performing computations for the vertical lines on the screen.

What exactly does art entail? The formation of figures and shapes, a fundamental comprehension of 2-D and 3-D, knowledge of spatial concepts, and the contribution of estimating, patterns, and measurement are all included in art. From the foregoing, it is clear that art and geometry have a close link. Geometrical forms such as the circle, triangle, square, mandala, and octagon are used to create shapes. Furthermore, the design and shape of frames have a significant impact on the content of paintings or sculptures. Not to mention that the principles of projective geometry are the foundation for most painting perspectives.

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In this article, we have understood the application of coordinate geometry in our day-to-day life. We have discussed how coordinate geometry is useful in our life and have discussed various examples of coordinate geometry which will help us to visualise the concept and ultimately help in a deeper understanding of the topic.

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Frequently asked questions

Get answers to the most common queries related to the CAT Examination Preparation.

What is the use of coordinate geometry in physics:-

What is the use of coordinate geometry in engineering:-, find the equation for the line with a 3 slope and a – 4 y intercept., determine the equation of a straight line passing through (2, 3) and parallel to the line 3x + 2y + 4 = 0. a.3y=2x+5 b.3y-2x-5=0 c.. 3y=5x-2 d. y=5/3x-2 explanation and answer, calculate the distance between p (2, 3) and the x-axis..

Answer: Analytic geometry, often known as coordinate geometry or Cartesian geometry in classical mathematics, is the study of geometry using a coordinate system. Synthetic geometry is the polar opposite. In addition to physics and engineering, analytic geometry is employed in aviation, rocketry, space research, and spaceflight.

Answer: Coordinate geometry is required to provide a link between algebra and geometry through the use of line and curve graphs. It is an important part of mathematics that helps us locate points on a plane. It also has several applications in trigonometry, calculus, dimensional geometry, and other sciences. 

Answer: A. 2x – 3 = y

Explanation and Answer

Option C is correct.

Suppose m = 3 and c = -4. We get y = 3x – 4 by substituting numbers in y = mx + c.

Option C is thus the solution.

Solution :  B

3x + 2y + 4 = 0 or y = -3x / 2 – 2 is the supplied line.

Slope = 2 / 3 for every line perpendicular to it

As a result, the equation of a line passing through (2, 3) and having a slope of 2 / 3 is (y – 3) = 2 / 3 (x – 2)

2x – 4 = 3y – 9

 Solution: We’re aware of this.

(x, y) = (2, 3) is a point in the first quadrant of the Cartesian plane.

x = Distance perpendicular to the y-axis

y = Distance perpendicular to the x-axis

As a result, the perpendicular distance between the x-axis and the y coordinate is 3.

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Steve Nadis

What Came Before the Big Bang?

The original version of this story appeared in Quanta Magazine .

About 13.8 billion years ago, the entire cosmos consisted of a tiny, hot, dense ball of energy that suddenly exploded.

That’s how everything began, according to the standard scientific story of the Big Bang, a theory that first took shape in the 1920s. The story has been refined over the decades, most notably in the 1980s, when many cosmologists came to believe that in its first moments, the universe went through a brief period of extraordinarily fast expansion called inflation before settling into a lower gear.

That brief period is thought to have been caused by a peculiar form of high-energy matter that throws gravity into reverse, “inflating” the fabric of the universe exponentially quickly and causing it to grow by a factor of a million billion billion in less than a billionth of a billionth of a billionth of a second. Inflation explains why the universe appears to be so smooth and homogeneous when astronomers examine it at large scales.

But if inflation is responsible for all that can be seen today, that raises the question: What, if anything, came before?

No experiment has yet been devised that can observe what happened before inflation. However, mathematicians can sketch out some possible scenarios. The strategy is to apply Einstein’s general theory of relativity—a theory that equates gravity with the curvature of space-time—as far back into time as it can go.

That’s the hope of three researchers: Ghazal Geshnizjani of the Perimeter Institute, Eric Ling of the University of Copenhagen, and Jerome Quintin of the University of Waterloo. The trio recently published a paper in the Journal of High Energy Physics in which, Ling said, “We mathematically showed that there might be a way to see beyond our universe.”

Working together with Jerome Quintin and Eric Ling, Ghazal Geshnizjani of the Perimeter Institute examined ways in which space-time might be extended beyond the Big Bang.

By Kelly Clancy

By Jaina Grey

By David Gilbert

By Kate Knibbs

Robert Brandenberger, a physicist at McGill University who was not involved with the study, said the new paper “sets a new standard of rigor for the analysis” of the mathematics of the beginning of time. In some cases, what appears at first to be a singularity—a point in space-time where mathematical descriptions lose their meaning—may in fact be an illusion.

A Taxonomy of Singularities

The central issue confronting Geshnizjani, Ling, and Quintin is whether there is a point prior to inflation at which the laws of gravity break down in a singularity. The simplest example of a mathematical singularity is what happens to the function 1/ x as x approaches zero. The function takes a number x as an input, and outputs another number. As x gets smaller and smaller, 1/ x gets larger and larger, approaching infinity. If x is zero, the function is no longer well defined: It can’t be relied upon as a description of reality.

“We mathematically showed that there might be a way to see beyond our universe,” said Eric Ling of the University of Copenhagen.

Sometimes, however, mathematicians can get around a singularity. For example, consider the prime meridian, which passes through Greenwich, England, at longitude zero. If you had a function of 1/longitude, it would go berserk in Greenwich. But there’s not actually anything physically special about suburban London: You could easily redefine zero longitude to pass through some other place on Earth, and then your function would behave perfectly normally when approaching the Royal Observatory in Greenwich.

Something similar happens at the boundary of mathematical models of black holes. The equations that describe spherical nonrotating black holes, worked out by the physicist Karl Schwarzschild in 1916, have a term whose denominator goes to zero at the event horizon of the black hole—the surface surrounding a black hole beyond which nothing can escape. That led physicists to believe that the event horizon was a physical singularity. But eight years later the astronomer Arthur Eddington showed that if a different set of coordinates is used, the singularity disappears. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity, which only arises because of the choice of coordinates.

At a black hole’s center, by contrast, the density and curvature go to infinity in a way that can’t be eliminated by using a different coordinate system. The laws of general relativity start spewing out gibberish. This is called a curvature singularity. It implies that something is taking place that’s beyond the ability of current physical and mathematical theories to describe.

Geshnizjani, Ling, and Quintin studied whether the onset of the Big Bang is more like the center of a black hole, or more like an event horizon. Their investigation builds upon a theorem proved in 2003 by Arvind Borde, Alan Guth (one of the first people to propose the idea of inflation), and Alexander Vilenkin. This theorem, known by the authors’ initials as BGV, says that inflation must have had a beginning—it can’t have been going on ceaselessly into the past. There must have been a singularity to kick things off. BGV establishes the existence of this singularity, without saying what kind of singularity it is.

As Quintin puts it, he and his colleagues have worked to figure out if that singularity is a brick wall—a curvature singularity—or a curtain that can be pulled back—a coordinate singularity. Eric Woolgar, a mathematician at the University of Alberta who was not involved in the study, said that it clarifies our picture of the Big Bang singularity. “They can say whether the curvature is infinite at the initial singularity or whether the singularity is milder, which might allow us to extend our model of the universe to times before the Big Bang.”

“Light rays can actually go through the boundary,” said Jerome Quintin of the University of Waterloo.

To classify possible pre-inflationary scenarios, the three researchers used a parameter called the scale factor that describes how the distance between objects has changed over time as the universe expands. By definition, the Big Bang is the time when the scale factor was zero—everything was squeezed into a dimensionless point.

During inflation, the scale factor increased with exponential speed. Before inflation, the scale factor could have varied in any number of ways. The new paper provides a taxonomy of singularities for different scale-factor scenarios. “We show that under certain conditions, the scale factor will produce a curvature singularity, and under other conditions it does not,” Ling said.

Researchers already knew that in a universe with so-called dark energy , but without matter, the start of inflation identified in the BGV theorem is a coordinate singularity that can be eliminated. But the real universe has matter, of course. Might mathematical tricks also make it possible to get around its singularity? The researchers showed that if the amount of matter is negligible compared to the amount of dark energy, then the singularity can be eliminated. “Light rays can actually go through the boundary,” Quintin said. “And in that sense, you can see beyond the boundary; it’s not like a brick wall.” The universe’s history would extend beyond the Big Bang.

However, cosmologists think that the early universe had more matter than energy. In this case, the new work shows that the BGV singularity would be a real physical curvature singularity, at which the laws of gravity stop making sense.

A singularity hints at the fact that general relativity can’t be a complete description of the basic rules of physics. Efforts to form such a description, which would require reconciling general relativity with quantum mechanics, are ongoing. Ling said he sees the new paper as a stepping stone to such a theory. In order to make sense of the universe at the highest energy levels, he said, “we first need to understand classical physics as well as we can.”

Original story reprinted with permission from Quanta Magazine , an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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Poland + 1 more

Grand Bargain Localization Commitments (Poland Case Study) June 2024

• NGO Forum - Razem

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Grand Bargain Localization Commitments (Poland Case Study) by Groupe URD for NGO Forum “Razem”

Financing Partners: CARE Poland, Oxfam Foundation in Poland, Norwegian Refugee Council in Poland, Contributing Partners: Plan International Poland, Save the Children International (Poland)

Steering Committee composed by representatives of: Foundation Ukraine, Migration Consortium and Mudita Association, CARE Poland, Oxfam Foundation in Poland, Norwegian Refugee Council in Poland

Operator: Polish Humanitarian Action

Authors, Groupe URD: Dominika Michalak, Véronique De Geoffroy, Rana Gabi, Elie Keldani, Karina Melnytska

REPORT SUMMARY:

Within the first month of Russia's full-scale invasion of Ukraine on 24 February 2022, over two million refugees crossed into Poland. Many continued to other European countries or overseas, but a year later, over a million refugees from Ukraine were registered for Temporary Protection in Poland. In 2024, refugees are still arriving in Poland from Ukraine, and the number of registered refugees remains similar to the previous year. This report evaluates the involvement of international humanitarian actors in response to the challenges related to this forced migration movement, as compared against the Grand Bargain Localization Commitments .

Its aim is to support the common work on better relationships between international actors, such as INGOs and UN agencies, and their local partners in Poland, as well as to contribute to the global localization debate.

The research employs the NEAR Localisation Performance Measurement Framework but also draws directly on analytical categories from the research material, i.e., on how the research participants understood the localization dynamics. The data were collected between March and May 2024. It provides information on the NGO landscape before the 24th of February 2022 and the development of the response up to the beginning of May 2024. In total**, 85 persons from 55 organizations took part in the research and 6 INGOs shared their partnership agreement templates for our analysis**.

Overall, the evaluation of the response dynamics against the Grand Bargain Commitments (as operationalized by the NEAR framework) showed positive results. Adherence to the commitments was highest with regard to financing: in the case of Poland, the main strengths of the localization processes were the high availability of direct funding , the high share of overhead costs covered by INGOs, and the good availability of financial support for organizational development. Capacity sharing and the quality of partnerships are areas where improvement was expected by all the actors involved. Capacity development often focused on facilitating local actors’ adaptation to the humanitarian system and rarely assumed the character of capacity sharing. Excessive formalization of partnerships after the emergency phase of the response, often combined with inconsistencies related to these formal requirements, meant office work overshadowed matters related to field challenges and impacted the quality of partnerships.

The report identifies 13 barriers and 10 enablers of localization in Poland .

THE BARRIERS include proximity bias in identification of projects’ participants and their needs; decontextualized character of humanitarian models, standards and commitments; no shared definition of success between local and international actors; excessive administrative burden on local actors; insufficient awareness raising about the key characteristics of the humanitarian cycle among L/NNGOs; high rotation of INGO and UN employees and language barriers.

THE ENABLERS include abundant direct funding available, including funding covering overhead costs and organizational development; availability of non-competitive funding; reliability of L/NNGOs as partners; horizontal networks facilitating cooperation between organizations sharing interests and facing similar challenges; local actors with adequate experience assuming the role of intermediaries (e.g., engaging in re-granting or facilitating coordination); high social legitimacy of providing support to Ukrainian refugees; authentic work on improving partnerships; acknowledgment of local expertise and capacity. We stress that localization is largely a matter of balance: some of the enablers, when applied without monitoring or on too wide a scale, can become barriers.

HIGHLIGHTED RECCOMENDATIONS:

- Strengthening networks of cooperation , i.e., horizontal networks between local actors and similar cooperation ties between international actors, has been instrumental in overcoming some of the localization barriers and is worth considering at other sites of humanitarian intervention.

- Capacity sharing instead of capacity building to acknowledge and properly employ the expertise of local actors is recommended in any context.

- Ensuring more equity in contracting , especially with regard to prioritizing the local law as the governing law and in terms of termination is recommended, especially for contexts where the rule of law is sufficient to ensure both sides a fair trial. Ensuring availability of contracts in the local language is recommended as general good practice.

- The report also concludes that reconnecting with social movements is beneficial to international humanitarian actors wherever these movements address questions at the heart of humanitarian ethics, such as human rights, civil participation or protection from violence.

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• Published: 25 June 2024

Tipping point in ice-sheet grounding-zone melting due to ocean water intrusion

• Alexander T. Bradley   ORCID: orcid.org/0000-0001-8381-5317 1 &
• Ian J. Hewitt   ORCID: orcid.org/0000-0002-9167-6481 2  

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• Climate and Earth system modelling
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Marine ice sheets are highly sensitive to submarine melting in their grounding zones, where they transition between grounded and floating ice. Recently published studies of the complex hydrography of grounding zones suggest that warm ocean water can intrude large distances beneath the ice sheet, with dramatic consequences for ice dynamics. Here we develop a model to capture the feedback between intruded ocean water, the melting it induces and the resulting changes in ice geometry. We reveal a sensitive dependence of the grounding-zone dynamics on this feedback: as the grounding zone widens in response to melting, both temperature and flow velocity in the region increase, further enhancing melting. We find that increases in ocean temperature can lead to a tipping point being passed, beyond which ocean water intrudes in an unbounded manner beneath the ice sheet, via a process of runaway melting. Additionally, this tipping point may not be easily detected with early warning indicators. Although completely unbounded intrusions are not expected in practice, this suggests a mechanism for dramatic changes in grounding-zone behaviour, which are not currently included in ice-sheet models. We consider the susceptibility of present-day Antarctic grounding zones to this process, finding that both warm and cold water cavity ice shelves may be vulnerable. Our results point towards a stronger sensitivity of ice-sheet melting, and thus higher sea-level-rise contribution in a warming climate, than has been previously understood.

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Heterogeneous melting near the Thwaites Glacier grounding line

Iceberg melting substantially modifies oceanic heat flux towards a major Greenlandic tidewater glacier

Suppressed basal melting in the eastern Thwaites Glacier grounding zone

There is growing evidence suggesting that ice-sheet models lack representation of important physical processes driving ice sheet retreat (for example, refs. 1 , 2 , 3 , 4 , 5 , 6 , 7 ), rendering their projections of sea-level rise less sensitive to climatic changes than they should be. Point-wise observations 1 , 2 suggest that ice-shelf basal melt rates are considerably smaller than those typically required by models to reproduce observed retreat rates. Moreover, ice-sheet models systematically underestimate recent ice loss 7 and struggle to reproduce observationally constrained sea-level highstands from previous interglacials 3 , 4 . Palaeoclimate ice-sheet reconstructions have largely been able to reproduce low-end estimates only when mechanisms to boost sensitivity to climatic forcing are invoked 8 , 9 , 10 . Recently, evidence from diverse sources has emerged suggesting that relatively warm ocean water can intrude long distances upstream of ice-shelf grounding lines 5 , 6 , 11 , 12 , 13 , 14 ; such long intrusions have dramatic consequences for sea-level-rise contributions from ice sheets 5 , 15 , making them a candidate mechanism to reconcile modelled and observed sea-level rise. Here we investigate how previously ignored feedbacks between melting and the confining ice geometry make this intrusion mechanism even more powerful.

Sea levels during previous interglacials can be considered analogues for future sea-level rise under anthropogenic influence 16 . During the Pliocene ( ∼ 3 million years before present), CO 2 levels were similar to present day 17 and temperatures 2–3 °C above pre-industrial levels 18 , but the global mean sea level (GMSL) was 6–40 m above present-day levels 4 . Such sea-level rise is only possible with a sizeable contribution from the Antarctic ice sheet 16 . Whereas proxy records suggest that such Antarctic ice loss occurred during previous interglacials (for example, refs. 19 , 20 , 21 ), ice-sheet models struggle to reproduce the corresponding ice-sheet retreat (for example, ref. 22 ).

Several palaeoclimate simulations 8 , 10 , 22 that attain low-end Pliocene GMSL estimates have gained interest because of their pessimistic Antarctic ice loss projections. To reconcile observational constraints, these models incorporate processes that increase their sensitivity to past climatic change, naturally rendering them more sensitive to future anthropogenic warming. Refs. 8 , 10 achieve low-end Pliocene GMSL by introducing a cliff-collapse mechanism, whereby sufficiently tall ice cliffs collapse, prompting rapid inland ice front retreat. However, these simulations have been questioned on both physical 23 , 24 and statistical 25 grounds. Ref. 22 obtained similar Pliocene GMSL using a model whose increased sensitivity results from a parameterization of basal melting in grounding zones, where grounded ice transitions into a floating ice shelf (Fig. 1 ). Flow of grounded ice is particularly sensitive to grounding-zone melting 15 , 26 , 27 , 28 , because melt-induced thinning there both reduces basal drag and provides a thinning perturbation that propagates through the shelf, reducing buttressing.

The grounding zone (dashed circle in top left) of marine-terminating ice sheets features networks of tunnels, channels and porous sediments through which water moves (centre). Layered intrusion models (cross section at right) consider this region as a two-layer system: freshwater, at the local freezing temperature T f , enters the zone with average velocity U ∞ , where it meets warm ocean water of temperature T O and salinity S O . Here V is the velocity of ice above the channel, H ∞ is the characteristic vertical length scale of the upstream subglacial network and θ is the local angle of the seabed. The two-dimensional model should be taken to represent an along-grounding-zone average of the complex three-dimensional drainage system in the centre panel; in particular, there will be areas where the ice remains in contact with the bed (that is, the model does not assume the ice is floating everywhere).

Specifically, the model in ref. 22 interpolates melting across model grid cells either side of the traditional ‘grounding line’, where the ice is at the hydrostatic floatation thickness. Although this respects the fact that areas between grid cells around the grounding line may be exposed to warm ocean water, it has little physical basis beyond; in practice, grounding zones are highly complex regions, with myriad features including local topographic highs (for example, ref. 29 ), porous till layers (for example, ref. 30 ) and channels on multiple lengthscales (for example, ref. 31 ) (Fig. 1 ). Freshwater is delivered to the ocean from the upstream grounded ice through this region; where freshwater meets the relatively warm, salty ocean water, the lower density freshwater rises, permitting the warm water to intrude upstream of the grounding line 11 . There is growing evidence of warm water intrusions from diverse sources including surface observations 12 , 13 , satellite data 6 , 14 and ice-shelf basal features 32 .

Recent near-grounding-zone observations 1 , 2 confirm that warm ocean water can reach cavity extremities. However, observations are unable to probe regions far upstream of the grounding line. A lack of direct observations beneath grounded ice sheets, combined with their importance in large ice-sheet models, means that models of the grounding zone are essential. Recently published grounding-zone models 5 , 11 treat the region as a porous, two-layer system with cold, fresh subglacial discharge overlaying warm, saline ocean water. These models predict that warm water can intrude significant distances (up to kilometres) upstream of ice-sheet grounding lines 5 , delivering excess heat for sub-ice melting. Ice-sheet models including the intrusion mechanism show that ice loss is highly sensitive to this intrusion length 5 , with kilometre length intrusions potentially doubling rates of ice loss. Grounding-zone melting via warm water intrusion has, therefore, been suggested as a physically based mechanism for explaining ice-sheet retreat during past warm periods 5 .

Crucially, however, existing models lack feedbacks between melting and the confining ice geometry. As grounding zones widen in response to melting, both primary factors controlling melt rates—the amount of warm water entering the grounding zone and flow speeds within it—will be affected. In particular, this feedback may strongly influence the distance warm water is able to intrude beneath ice sheets, with significant implications for ice dynamics. References 5 , 11 demonstrated that with no feedback between melting and geometry, the distance warm water can intrude depends on the slope of the ice base and can become infinite if the slope is sufficiently steep. Here we show that with the melt-geometry feedback included, the system exhibits a tipping-point behaviour as the parameters controlling the melt rate are varied, causing unbounded intrusions to develop even on a flat or down-sloping bed; this behaviour can be triggered by changes in external forcing, such as ocean temperatures.

Melting causes enhanced warm water intrusion

To understand how the melt feedback affects grounding-zone behaviour, and, in particular, the distance warm water is able to intrude, we coupled the layered intrusion model of refs. 5 , 11 with a common melt-rate model accounting for the dependence of melting on both temperature and flow velocity adjacent to the ice (Methods).

The modelled grounding-zone behaviour depends on four fundamental, dimensionless parameters ( Supplementary Information ):

Here U ∞ and H ∞ are a characteristic flow velocity and thickness of the upstream hydrological network (Fig. 1 ); V is the grounding-zone ice velocity; St is the Stanton number, the ratio between the thermal flux into the ice–ocean interface and the thermal capacity of the water, which effectively parametrizes exchange across a boundary layer at the ice–ocean interface 33 ( Supplementary Information ); c is the specific heat capacity of water; c d is a cross-sectional average drag coefficient between the water and the channel; c is the specific heat capacity of ocean water; \({{\Delta}}T={T}_\mathrm{O}+{{\varGamma}}{{{\mathcal{S}}}}_\mathrm{O}-{T}_\mathrm{D}\) is the thermal forcing, with T O the ocean temperature, \({{{{\mathcal{S}}}}}_\mathrm{O}\) the ocean salinity, Γ the freezing point slope with salinity and T D the local freshwater freezing temperature; \({\mathcal{L}}\) is the latent heat of fusion of seawater; θ is the local grounding-zone slope, assumed constant ( Supplementary Information ); \({g}^{\,{\prime} }=g(\,{\rho }_{0}-{\rho }_\mathrm{w})/{\rho }_\mathrm{w}\) is the reduced gravity, with ρ 0 a reference density and ρ w the local water density; and c i is a cross-sectional average drag coefficient between the two layers. A full model description, including a discussion of underlying assumptions, can be found in the Supplementary Information .

These parameters capture the complex ice–ocean interactions that occur in grounding zones: M (equation ( 1a )) is a dimensionless melt rate, describing the competing effects of increasing ocean temperature (increasing Δ T ) in promoting enhanced melting, and increased ice advection (increasing V ) in replacing this ice; S (equation ( 1b )) is a dimensionless bedslope, with positive (negative, respectively) S corresponding to retrograde (prograde) bedslopes—upward (downward) sloping in the direction of ice flow; F (equation ( 1c )) is the upstream Froude number, which describes the upstream hydrological network efficiency: efficient networks, with fast flow (high U ∞ ) through narrow confines (low H ∞ ) correspond to large F , whereas in inefficient networks (low F ), meltwater is transported slowly through wide channels; C (equation ( 1d )) is a dimensionless interfacial drag coefficient, describing the relative importance of drag between the two water layers and between the water and solid (ice/bed) boundaries.

Figure 2 shows how the intrusion distance, grounding-zone geometry, thermal driving and flow velocity change as the geometry evolves from an initially flat ice base and no seabed slope in two cases with similar ocean conditions: one with Δ T  = 2.3 °C and the other with Δ T  = 2.5°C (we consider the case of a variable bedslope later). Melting is concentrated at the channel entrance, where the flow velocity and thermal driving are highest (Fig. 2c,d,g,h ) and reduces with distance into the channel. As melting proceeds, the grounding-zone widens (Fig. 2b,f ), permitting more warm water to enter, increasing the average temperature (Fig. 2c,g ), and reductions in drag result in higher flow velocities (Fig. 2d,h ); these work in tandem to promote enhanced melting. When ice is replaced by advection sufficiently quickly (left panels of Fig. 2 ), the system reaches a steady state with melting balancing ice advection and drag balancing the gravitational force that results from a titled warm–cold interface. This equilibrium is reached on a timescale of days (Fig. 2a ). However, for marginally higher ocean temperatures, ice advection cannot balance melting and the grounding zone continually widens (Fig. 2e,f ). The feedback between geometry, hydrology and melting results in runaway warm water intrusion. This system displays a tipping-point-like behaviour: a small change in the ocean temperature (and thus parameter M ) results in a threshold being passed, across which a dramatic change in the modelled final intrusion length L occurs, from being bounded (Fig. 2a ) to being unbounded (Fig. 2e ). The timescale on which the grounding zone responds to melting is much shorter than that on which the grounded ice thickness responds to perturbations in melting; it is therefore reasonable to consider the late time behaviour, and we henceforth refer to the final intrusion length L as the intrusion length. The large increase in L , and thus melting beneath a large section of the grounded ice sheet, would have dramatic implications for the dynamics of a marine-terminating ice sheet 5 , 15 . Note, however, that we do not expect intrusions to penetrate indefinitely in practice, because processes not included in our model will play a role on long lengthscales and potentially stabilize the intrusion (see below).

a , e , Temporal evolution of the intrusion distance—the greatest upstream extent of the warm layer—for a thermal forcing of Δ T  = 2.3 °C (left) and Δ T  = 2.5 °C (right). For Δ T  = 2.3 °C, the intrusion distance tends towards a bounded value ( L  ≈ 110 m), indicated by the black dashed line, whereas for Δ T  = 2.5 °C, the intrusion becomes unbounded ( L  →  ∞ ) (note the different ordinate scales between the left and right panels). Translucent points in e show the data in a . b , f , Evolution of grounding-zone channel surface (solid curves) and warm–cold interface (dashed curves). Snapshots are shown at times 1, 5, 10, 50 and 100 days after initialization with a flat channel (the same initial condition is used in both cases). y is the distance to the seabed, which forms the base of the channel. Filled points indicate the intrusion distance and correspond to points shown in a as indicated by Roman numerals. In b , the dashed black line indicates the steady state intrusion distance L . c , d , g , h , Profiles of thermal driving ( c , g ) and ice-adjacent flow velocity ( d , h ) corresponding to the snapshots shown in ( b , f ). Solutions here correspond to a flat bed ( S  = 0), a fairly inefficient drainage system ( F  = 0.25) and C  = 0.1.

Grounding-zone melt as a generic tipping point

This tipping point is generic: for any hydrological network efficiency F , the intrusion length increases with the melt parameter M (Fig. 3 ) and there is a critical M above which the intrusion becomes unbounded (solid line in Fig. 3 ). Equivalently, for any M , there is a critical F , named F c , below which the intrusion becomes unbounded. Although less efficient networks (lower F ) correspond to lower upstream flow velocities (lower U ∞ , see equations ( 1a,b ) and ( 1c )), this is outweighed by reduced drag between the layers, resulting in higher flow speeds adjacent to the ice and a thicker warm water layer, both of which promote increased melting. The critical hydrological network efficiency F c is increasing with M (Fig. 3 ): higher melting is required to cross the tipping point for more efficient subglacial networks. This suggests that increases in the flow of water beneath ice sheets may act as a stabilizing control on their dynamics via reduced grounding-zone melting, contrasting the common belief that subglacial flow predominantly enhances ice loss via reduced basal friction 34 .

Regime diagram of dimensionless intrusion distance L /( H ∞ / c d ) as a function of upstream Froude number F and dimensionless melt rate M for C  = 0.1 and S  = 0, corresponding to a flat base. Coloured lines indicate the bounded–unbounded intrusion length transition for different values of C , as indicated. The red line indicates schematically the transition between the two values of M used in Fig. 2 .

The location of the transition between bounded and unbounded intrusions is relatively insensitive to the dimensionless drag coefficient C (Fig. 3 ). This provides support for our use of a two-dimensional model, because C encodes heterogeneity in the along-grounding-zone direction.

It is interesting to note that when the intrusion is bounded, the intrusion length L is fairly insensitive to the melt parameter M (Fig. 3 ). This suggests that early warning indicators 35 , which might indicate that a marine ice sheet is approaching such a grounding-zone melt tipping point as (say) the ocean temperature increases, may be hard to detect: an increase in the intrusion length would not appear as a detectable signal in ice dynamical observations until the tipping point is passed, propagating uncertainty into sea-level-rise projections 36 .

Widespread susceptibility to the tipping point in melting

Intrusion of dense seawater is ultimately gravity driven and therefore strongly modified by the grounding-zone bedslope. Retrograde bedslopes ( S  > 0) result in enhanced intrusion, whereas prograde bedslopes ( S  < 0) result in reduced intrusion, compared to a flat bed (Fig. 4a and Supplementary Fig. 8 ). Unbounded intrusion can occur with no melt feedback (equivalent to M  → 0), provided that the slope is sufficiently retrograde, under conditions described by ref. 5 . However, with melt feedbacks, unbounded intrusion can occur for any bedslope, including prograde (Fig. 4a ): when melting is sufficiently strong, the widening of the warm layer accompanying channel opening creates a gravitational driving force that overcomes the retarding gravitational effect from the bedslope, which is otherwise unfavourable for intrusion. The critical bedslope, S c , above which unbounded intrusion occurs, reduces with M (Fig. 4a ). Over large regions of parameter space, S c is negative (Supplementary Fig. 7 ): that is, our model suggests that unbounded intrusion may occur even on prograde bedslopes and particularly so for inefficient hydrological networks (low F ; Supplementary Fig. 7 ).

a , Dimensionless intrusion distance L /( H ∞ / c d ) as a function of dimensionless bedslope S for different M , as indicated by the colour bar. Filled circles indicate the critical slope S c , beyond which the intrusion becomes unbounded. The black dashed line indicates S  = 0, which delineates pro- ( S  < 0) and retrograde ( S  > 0) bed slopes. Data shown here correspond to F  = 0.5 and C  = 0.1. b , Curves of critical upstream Froude number F c , below which unbounded intrusion occurs, as a function of dimensionless slope S for different M , as indicated by the colour bar in a . c , Map of critical upstream Froude upstream number F c . Points indicate ice shelves in Antarctica (PIG = Pine Island Glacier, PSK = Pope–Smith–Kohler), determined from observations (see main text). Grey contours correspond to F c  = 0.3 and F c  = 0.7. Ellipses indicate errors in distributions used to generate values of S and M . Points on the inset indicate locations of those ice shelves in the main panel with corresponding colours.

Marine ice sheets grounded on retrograde bedslopes may be susceptible to the marine ice-sheet instability (MISI) (for example, refs. 37 , 38 , 39 ). Retreat of the West Antarctic ice sheet, many areas of which have grounding zones on retrograde bed slopes 40 , may be strongly controlled by MISI 41 . Our modelling suggests warm water intrusion is most likely on retrograde bedslopes, potentially enhancing MISI. Conversely, it is commonly assumed that prograde grounding lines are stable 38 , 42 ; our results suggest prograde grounding zones can also host substantial intrusions and have the possibility for a switch in behaviour as ocean temperatures change, potentially questioning their assumed stability.

In practice, hydrological networks beneath ice sheets are poorly constrained, making it difficult to determine their efficiency and thus F . However, M and S can be determined from observations (below). Therefore, to place our results in a present-day context, we consider the critical hydrological network efficiency, F c , which is shown as a function of M and S in Fig. 4c (as before, unbounded intrusion occurs for F  <  F c ). Assuming a uniform hydrological network efficiency for all ice shelves, F c is a proxy for susceptibility to passing the tipping point: regions of parameters space with darker (lighter, respectively) colours in Fig. 4c are more (less) susceptible to unbounded intrusions. Locations of key Antarctic ice shelves on this map are determined using the median of observations of grounding-line velocity 43 , basal slope 40 , ocean thermal forcing 44 and literature standard values for other parameters (Methods).

Despite a large spread in the data, particularly in the grounding-line slope, we find that on average, the rapidly accelerating 45 and thinning 46 Thwaites Glacier is the least susceptible of those ice shelves considered. This is perhaps surprising, given its high ocean forcing (Supplementary Fig. 5 ); however, its high grounding-line velocities (Supplementary Fig. 6 ) overcompensate for this melting potential. This highlights the stabilizing potential of high grounding-line velocities to warm water intrusion. However, Thwaites may be particularly sensitive to future changes: ice shelves corresponding to smaller M are more sensitive to changes in grounding-line slope (Fig. 4b ), which Thwaites may be exposed to as it maintains its present retreat 40 .

The Filchner and Amery ice shelves also have low susceptibility (Fig. 4c ). Compared to Thwaites, however, their low susceptibility results from low ocean forcing and strongly prograde bedslopes (Supplementary Figs. 4 – 6 ), respectively, rather than high grounding-line velocities. Pine Island, on the other hand, has high susceptibility; like Thwaites, it has high grounding-line velocities, but its grounding-line slope is higher (more retrograde), on average, and thus more favourable for intrusion (Supplementary Fig. 4 ). Pine Island is currently Antarctica’s largest contributor to sea-level rise; its high susceptibility to grounding-zone melting may represent another factor, in addition to a highly damaged ice shelf 47 and predicted future increases in ice-shelf melting 48 , 49 , which promotes its ongoing retreat. Both Getz and Larsen have similar susceptibility to Pine Island; although Getz has similar thermal forcing, its lower grounding-line velocities (higher M ) are balanced by reduced grounding-line slopes (lower S ). Larsen has high susceptibility owing to its low flow speeds (large M ): ice is not advected quickly enough to replace that removed by melting. More generally, these examples highlight the complex interplay between ice velocity, ocean forcing and bedslope in controlling melt in grounding zones of marine ice sheets.

Our results do not provide a prediction of which Antarctic ice shelves currently experience warm water intrusions but rather indicates their potential for such and their relative susceptibility. We speculate, however, that properties of subglacial hydrological networks may be inferred from our results; for example, recent observations of high melt upstream of the Thwaites grounding line 14 suggests this area may be in the unbounded intrusion regime. According to our modelling, such behaviour would require an inefficient subglacial network (Fig. 4c ), which is consistent with observations of ponding in distributed canals beneath Thwaites 50 .

We stress that completely unbounded intrusions are not expected in practice but rather the possibility of large, rapid, increases in intrusion distance and thus melting; on longer lengthscales, processes such as bedslope variations and melt feedbacks on channel temperature may suppress intrusion ( Supplementary Information ), and along-grounding-zone variations, not included in our model, may play an important role. Our model does not provide a prediction of the ice-dynamic response to unbounded intrusion, which requires a coupled ice-hydrology model to determine; increases in melting may lead to ice acceleration (increasing V ), potentially stabilizing the intrusion (effectively reducing M ). Finally, our model does not include tides. Grounding-zone characteristics may vary substantially over tidal cycles (for example, refs. 51 , 52 ), potentially affecting intrusion. As an ice shelf is raised in response to tides, water is forced into the grounding zone and evacuated as the ice shelf lowers. Given that grounding lines can migrate long (up to kilometre) distances over diurnal tidal cycles 51 , 53 , this has the possibility to create rapid flow in the grounding zone. In addition, the associated tidal flexure may lead to a modification of the grounding-zone geometry, potentially feeding back on intrusion, which is sensitive to the characteristic thickness of the subglacial environment 5 . Tidal currents may also modulate near-grounding-zone ocean circulation 2 , potentially altering flow boundary conditions on grounding zones and thus intrusion.

A complete treatment of grounding-zone flow on tidal timescales is beyond the scope of this work. However, when supplemented with a simple parameterization of tidal flow ( Supplementary Information ), our model still displays the tipping-point behaviour, with the location of the tipping point (in parameter space) modulated by the tidal amplitude (Supplementary Fig. 10 ). We find that tidal fluctuations can significantly enhance intrusion. This provides further motivation to better understand tidal influences on grounding zones, and, more generally, to develop high-resolution models of grounding zones and better constrain their characteristics via improved observations.

We have shown that feedbacks between subglacial water flow, melting and the confining ice geometry can result in increases in warm water intrusion into marine ice-sheet grounding zones, which would have implications for ice dynamics. In particular, we have identified a fundamental switch between bounded and unbounded warm water intrusions, occurring across a critical parameter threshold. The tipping point is generic: it exists for any marine-terminating ice sheet exposed to sufficiently warm ocean water, has sufficiently low grounding-line velocity or basal slopes or a sufficiently weak hydrological network. We have shown that the intrusion mechanism is stronger than previously understood, lending further credence to the theory that it is a physically based ‘sensitivity-boosting mechanism’ to reconcile the gap between observed and modelled sea-level rise in previous warm periods and the basal melt rates required to reproduce observed retreat. Current sea-level-rise projections for Antarctica and Greenland 54 are based on simulations that lack grounding-zone melting via intrusion and may therefore represent underestimates. Although our model is a simplification of the myriad complex processes occurring in grounding zones, the possibility of tipping points in grounding-zone melt and the universality of susceptible shelves warrants a continued research effort to better constrain grounding-zone processes both from models and observations.

Layered intrusion model

The layered intrusion model is identical to that of ref. 5 in the hard bed limit ( γ  → 0 in the nomenclature of ref. 5 ). This model builds upon that of ref. 11 and was verified experimentally therein. It is described in full detail in the Supplementary Information .

We couple the layered intrusion model to a simple model of melting,

where \(\dot{m}\) is the melt rate, \({\mathcal{L}}={3.35\times {10}^{5}}\;{\mathrm{J}\;{\rm{Kg}}^{-1}}\) is the latent heat of fusion of seawater, c  = 3.974 × 10 3 J Kg −1  °C −1 is the specific heat capacity of water and St is a combined Stanton number, which parametrizes combined exchange of salt and heat across a thermal boundary layer that forms on the ocean side of the ice–ocean interface 55 , 56 ( u * and τ are defined below). The Stanton number is fairly poorly constrained in general. In the results presented here, we take value St = 5.9 × 10 −4 ; this value is standard in the literature and was obtained from a fit to data obtained from beneath the Ronne ice shelf 57 .

Equation ( 3 ) results from the so-called ‘two-equation formulation’ of melting 55 in the limit of low diffusive heat flux (this is reasonable as freezing and internal temperatures of the ice are typically within a few degrees of one another 56 ) ( Supplementary Information ). The two-equation formulation is a simplification of the more detailed ‘three equation formulation’ 33 , 58 in which salt and heat exchange across the boundary layer at the ice-shelf base are considered separately rather than together as in the two-equation formulation. However, the two formulations have been shown to work equally well in several observational (for example, ref. 57 ) and numerical (for example, ref. 59 ) studies.

In equation ( 3 ), u * and τ are the velocity of the water adjacent to the ice–ocean interface and the thermal driving, respectively (the latter should not be confused with the thermal forcing Δ T ). We take u * to be the velocity of the fresh layer, which is the layer adjacent to the ice–ocean interface. The thermal driving is τ  =  T  −  T f , where T is the temperature adjacent to the ice–ocean interface (below) and \({T}_\mathrm{f}={T}_{\rm{ref}}+\lambda z-{{\varGamma}}{\mathcal{S}}\) the local freezing temperature, with T ref  = 8.32 × 10 −2  °C a reference temperature, λ  = 7.61 × 10 −4  °C m −1 the liquidus slope with depth, Γ  = 5.73 × 10 −2 the liquidus slope with salinity, \({\mathcal{S}}\) the local salinity and z the depth below sea level (more negative z corresponds to a greater depth) 60 .

We take a simple model for the channel temperature and salinity, assuming that the relevant temperature and salinity that drive melting are the depth-weighted average of the layer temperatures:

where ϕ is the column-wise proportion of the channel occupied by the freshwater layer ( Supplementary Information ), T D  = 0 and S D  = 0 are the temperature and salinity of the subglacial discharge layer, respectively, and T O and S O are the temperature and salinity of the warm ocean layer. The relations ( 4 ) and ( 5 ) capture the fact that the temperature and salnity in the channel increase with a greater proportion of warm, salty ocean water within it. Although entrainment between the two layers is not explicitly resolved, the column-wise averaging can be considered a simple proxy for mixing of the two layers. Observations also indicate that basal melting can occur where a cold fresh layer exists adjacent to an ice–ocean interface through double diffusive convection 52 , 61 .

Channel shape evolution

The dimensionless model equations (equations ( 19 ) and ( 20 ) in Supplementary Information ) are solved numerically in MATLAB. For a given channel shape, the layered intrusion equations are solved using the ODE15S routine. The equations are solved backwards from the downstream end of the channel, where we apply a perturbed boundary condition, setting the dimensionless freshwater layer thickness equal to \({\left[(1+\epsilon )F\right]}^{2/3}\) , where ϵ   ≪  1 ( Supplementary Information ). This perturbed boundary condition ensures that a singularity in the interfacial gradient is avoided at the downstream end of the channel 5 . In those results shown here, we use ϵ  = 10 −4 but verified that results are insensitive to this value, provided that the ϵ   ≪  1 condition holds. The intrusion equations are integrated backwards until either the freshwater layer occupies the entirety of the channel or the end of the numerical grid is reached (we use a sufficiently large numerical grid to ensure that the latter is only realized in the case of unbounded intrusion).

Having determined the interfacial shape, and thus velocity in the fresh layer and channel temperature, the melt rate is determined from equation ( 3 ). This melt rate is interpolated onto a regular grid with spacing d z (below) and the channel thickness timestepped according to the kinematic condition (equation ( 14 ) in the Supplementary Information ) using a first-order upwinding scheme 62 .

The numerical grid is made up of m blocks of n grid cells (giving a total number of grid cells of m  ×  n ); each block is of length L p  = 1 −  F 2 /4 − 3 F 2/3 /4, which is the intrusion distance in the limit of no interfacial drag ( C  = 0), a flat bed ( S  = 0) and no melting ( M  = 0) as described by ref. 5 ; the grid size is then d z  =  L p / n . In the results shown herein, we use n  = 100 and m  = 20, with the latter value being sufficiently large that the intrusion only reaches the end of the channel in the case of an unbounded intrusion.

Steady intrusion length

To determine the steady intrusion length L for a given set of parameters ( F ,  C ,  S ,  M ), we integrate the steady form of the coupled layered intrusion-melt equations (equations ( 24 ) and ( 25 ) in the Supplementary Information ) downstream from the nose of the wedge (where the freshwater layer occupies the width of the channel) using the ODE15S routine in MATLAB. At the nose, the problem is singular; to avoid this singularity, we linearize the problem about this point to determine the appropriate initial conditions ( Supplementary Information ). Solutions of this steady problem, which obtain the downstream boundary condition (dimensionless freshwater layer thickness =  F 2/3 ) in a finite distance correspond to true steady states; otherwise, no steady solution exists for this particular set of parameters, and the intrusion will be unbounded (Supplementary Fig. 2 ). In practice, we specify a finite end of the domain ℓ   ≫  1, and, if the solution does not obtain the downstream boundary condition before this point, we assume the intrusion is unbounded; in the results shown here, we take ℓ  = 10 5 , and results are insensitive to this value.

To determine the critical slope S c and critical Froude number F c shown in Fig. 4 , we apply a bisection method. The algorithm to do so is described fully in the Supplementary Information .

Parameter estimation

Values of parameters M and S for Antarctic ice shelves shown in Fig. 4 were determined from observations of ice velocity (for grounding-line velocity V ), thermal forcing (for Δ T ) and bedslopes (for θ ).

To determine grounding-line locations, we first obtain ice-shelf boundaries from Bedmachine V3 40 masks of ice-shelf location at 500 m resolution. Grid points within this mask corresponding to grounding-line and ice-shelf front positions were differentiated based on a floatation condition, relating to the floatation thickness ρ w / ρ i b , the ice thickness at which hydrostatic equilibrium is achieved, where ρ w  = 1,028.0 kg m −3 is the ocean density, ρ i  = 918.0 kg m −3 is the ice density and b is the bed elevation determined from Bedmachine V3 bed data 40 . Ice-shelf boundary points above 95% of floatation thickness were designated as grounding-line points, whereas the remaining points were designated as ice front points. Supplementary Fig. 9 shows grounding-line and ice front points for each of the ice shelves shown in Fig. 4 , indicating that this criterion does a good job at correctly identifying, and differentiating between, grounding-line and ice front points.

Ice velocities at grounding-line points were determined from NASA ITS_LIVE mosaics of 1985–2019 ice velocities 43 . We first put the data onto the same 500 m resolution grid used to determine grounding-line locations and then extract velocity components v  = ( v 1 ,  v 2 ) at these points (inset in Supplementary Fig. 9a ). Grounding-line ice velocities used to compute M are taken as the median of all grounding-line velocities within the individual ice shelf (Supplementary Fig. 6 ).

Grounding-line slopes were determined by first extracting bed data from Bedmachine V3 40 onto the 500 m grid. We then compute gradients of basal elevation,  ∇   b  = (∂ x b , ∂ y b ), where ∂ indicates a partial derivative, using second order finite differences. We then take the basal slope as the directional derivative in the direction of ice flow, that is

The value of \(\tan \theta\) used for each is shelf is then determined as the median of all grounding-line basal slopes for that particular shelf (Supplementary Fig. 4 ).

Thermal forcing is computed from maps of maximum thermal forcing between 200 m and 800 m in the water column from ref. 44 . For each ice front grid point on the 500 m grid, the thermal forcing associated with that point is computed as the mean of the maximum thermal forcing within a 1.5 km × 1.5 km square centred around the grid point; the thermal forcing of each ice shelf is then determined as the median over all well-defined ice front points in the shelf (Supplementary Fig. 5 ).

Having determined V , Δ T and \(\tan \theta\) from observations, M and S are computed using standard values from the literature. We take C d  = 10 −2 , which is consistent with a range of observations (for example, ref. 63 ), modelling (for example, ref. 48 ) and experiments (for example, ref. 64 ) of ice–ocean interactions, alongside \({\mathcal{L}}={3.35\times {10}^{5}}\;{\mathrm{J}\;{\rm{Kg}}^{-1}}\) , c  = 3.974 × 10 3 J Kg −1  °C −1 and St = 5.9 × 10 −4 as discussed above. The mean upstream flow velocity U ∞ is taken to be 1 cm s −1 , which is consistent with observations 65 , 66 and modelling 67 , 68 of subglacial flow beneath ice sheets.

Data availability

Data used to create the figures contained in this paper are available via Zenodo at https://doi.org/10.5281/zenodo.10895498 (ref. 69 ).

Code availability

Code to perform simulations and produce the figures contained in this paper are available via Zenodo at https://doi.org/10.5281/zenodo.10895498 (ref. 69 ).

Davis, P. E. et al. Suppressed basal melting in the eastern Thwaites Glacier grounding zone. Nature 614 , 479–485 (2023).

Article   CAS   Google Scholar  

Schmidt, B. E. et al. Heterogeneous melting near the Thwaites Glacier grounding line. Nature 614 , 471–478 (2023).

Dutton, A. & Lambeck, K. Ice volume and sea level during the last interglacial. Science 337 , 216–219 (2012).

Dumitru, O. A. et al. Constraints on global mean sea level during Pliocene warmth. Nature 574 , 233–236 (2019).

Robel, A. A., Wilson, E. & Seroussi, H. Layered seawater intrusion and melt under grounded ice. Cryosphere 16 , 451–469 (2022).

Article   Google Scholar  

Cirací, E. et al. Melt rates in the kilometer-size grounding zone of Petermann Glacier, Greenland, before and during a retreat. Proc. Natl Acad. Sci. USA 120 , e2220924120 (2023).

Aschwanden, A., Bartholomaus, T. C., Brinkerhoff, D. J. & Truffer, M. Brief communication: a roadmap towards credible projections of ice sheet contribution to sea level. Cryosphere 15 , 5705–5715 (2021).

DeConto, R. M. & Pollard, D. Contribution of Antarctica to past and future sea-level rise. Nature 531 , 591–597 (2016).

Golledge, N. R. et al. Global environmental consequences of twenty-first-century ice-sheet melt. Nature 566 , 65–72 (2019).

DeConto, R. M. et al. The Paris Climate Agreement and future sea-level rise from Antarctica. Nature 593 , 83–89 (2021).

Wilson, E. A., Wells, A. J., Hewitt, I. J. & Cenedese, C. The dynamics of a subglacial salt wedge. J. Fluid Mech. 895 , A20 (2020).

MacGregor, J. A., Anandakrishnan, S., Catania, G. A. & Winebrenner, D. P. The grounding zone of the Ross Ice Shelf, West Antarctica, from ice-penetrating radar. J. Glaciol. 57 , 917–928 (2011).

Horgan, H. J. et al. Estuaries beneath ice sheets. Geology 41 , 1159–1162 (2013).

Milillo, P. et al. Heterogeneous retreat and ice melt of Thwaites Glacier, West Antarctica. Sci. Adv. 5 , eaau3433 (2019).

Seroussi, H. & Morlighem, M. Representation of basal melting at the grounding line in ice flow models. Cryosphere 12 , 3085–3096 (2018).

Dutton, A. et al. Sea-level rise due to polar ice-sheet mass loss during past warm periods. Science 349 , aaa4019 (2015).

Seki, O. et al. Alkenone and boron-based pliocene p CO 2 records. Earth Planet. Sci. Lett. 292 , 201–211 (2010).

Haywood, A. M. et al. The Pliocene Model Intercomparison Project Phase 2: large-scale climate features and climate sensitivity. Clim 16 , 2095–2123 (2020).

Google Scholar  

Scherer, R. P. et al. Pleistocene collapse of the West Antarctic Ice Sheet. Science 281 , 82–85 (1998).

Hillenbrand, C.-D. et al. West Antarctic Ice Sheet retreat driven by Holocene warm water incursions. Nature 547 , 43–48 (2017).

Wilson, D. J. et al. Ice loss from the East Antarctic Ice Sheet during late Pleistocene interglacials. Nature 561 , 383–386 (2018).

Golledge, N. R. et al. Antarctic climate and ice-sheet configuration during the early Pliocene interglacial at 4.23 Ma. Clim 13 , 959–975 (2017).

Clerc, F., Minchew, B. M. & Behn, M. D. Marine ice cliff instability mitigated by slow removal of ice shelves. Geophys. Res. Lett. 46 , 12108–12116 (2019).

Bassis, J., Berg, B., Crawford, A. & Benn, D. Transition to marine ice cliff instability controlled by ice thickness gradients and velocity. Science 372 , 1342–1344 (2021).

Edwards, T. L. et al. Revisiting Antarctic ice loss due to marine ice-cliff instability. Nature 566 , 58–64 (2019).

Parizek, B. et al. Dynamic (in) stability of Thwaites Glacier, West Antarctica. J. Geophys. Res. Earth Surf. 118 , 638–655 (2013).

Arthern, R. J. & Williams, C. R. The sensitivity of West Antarctica to the submarine melting feedback. Geophys. Res. Lett. 44 , 2352–2359 (2017).

Reese, R., Gudmundsson, G. H., Levermann, A. & Winkelmann, R. The far reach of ice-shelf thinning in Antarctica. Nat. Clim. Change 8 , 53–57 (2018).

Graham, A. G. et al. Rapid retreat of Thwaites Glacier in the pre-satellite era. Nat. Geosci. 15 , 706–713 (2022).

Anandakrishnan, S., Catania, G. A., Alley, R. B. & Horgan, H. J. Discovery of till deposition at the grounding line of Whillans Ice Stream. Science 315 , 1835–1838 (2007).

Dow, C. F., Ross, N., Jeofry, H., Siu, K. & Siegert, M. J. Antarctic basal environment shaped by high-pressure flow through a subglacial river system. Nat. Geosci. 15 , 892–898 (2022).

Whiteford, A., Horgan, H., Leong, W. & Forbes, M. Melting and refreezing in an ice shelf basal channel at the grounding line of the Kamb Ice Stream, West Antarctica. J. Geophys. Res. Earth Surf. 127 , e2021JF006532 (2022).

Jenkins, A. Convection-driven melting near the grounding lines of ice shelves and tidewater glaciers. J. Phys. Oceanogr. 41 , 2279–2294 (2011).

Hewitt, I. Seasonal changes in ice sheet motion due to melt water lubrication. Earth Planet. Sci. Lett. 371 , 16–25 (2013).

Lenton, T. M. Early warning of climate tipping points. Nat. Clim. Change 1 , 201–209 (2011).

Robel, A. A., Seroussi, H. & Roe, G. H. Marine ice sheet instability amplifies and skews uncertainty in projections of future sea-level rise. Proc. Natl Acad. Sci. USA 116 , 14887–14892 (2019).

Weertman, J. Stability of the junction of an ice sheet and an ice shelf. J. Glaciol. 13 , 3–11 (1974).

Schoof, C. Ice sheet grounding line dynamics: steady states, stability, and hysteresis. J. Geophys. Res. Earth Surf. 112 , F03S28 (2007).

Katz, R. F. & Worster, M. G. Stability of ice-sheet grounding lines. Philos. Trans. R. Soc. A 466 , 1597–1620 (2010).

Morlighem, M. et al. Deep glacial troughs and stabilizing ridges unveiled beneath the margins of the Antarctic Ice Sheet. Nat. Geosci. 13 , 132–137 (2020).

Favier, L. et al. Retreat of Pine Island Glacier controlled by marine ice-sheet instability. Nat. Clim. Change 4 , 117–121 (2014).

Schoof, C. Marine ice sheet stability. J. Fluid Mech. 698 , 62–72 (2012).

Gardner, A. S. et al. Increased West Antarctic and unchanged East Antarctic ice discharge over the last 7 years. Cryosphere 12 , 521–547 (2018).

Adusumilli, S., Fricker, H. A., Medley, B., Padman, L. & Siegfried, M. R. Interannual variations in meltwater input to the Southern Ocean from Antarctic ice shelves. Nat. Geosci. 13 , 616–620 (2020).

Mouginot, J., Rignot, E. & Scheuchl, B. Sustained increase in ice discharge from the Amundsen Sea Embayment, West Antarctica, from 1973 to 2013. Geophys. Res. Lett. 41 , 1576–1584 (2014).

Pritchard, H. et al. Antarctic ice-sheet loss driven by basal melting of ice shelves. Nature 484 , 502–505 (2012).

Lhermitte, S. et al. Damage accelerates ice shelf instability and mass loss in Amundsen Sea Embayment. Proc. Natl Acad. Sci. USA 117 , 24735–24741 (2020).

Bradley, A. T., Bett, D., Dutrieux, P., De Rydt, J. & Holland, P. R. The influence of Pine Island Ice Shelf calving on basal melting. J. Geophys. Res. Oceans 127 , e2022JC018621 (2022).

Bradley, A. T., De Rydt, J., Bett, D. T., Dutrieux, P. & Holland, P. R. The ice dynamic and melting response of Pine Island Ice Shelf to calving. Ann. Glaciol. 63 , 111–115 (2022).

Schroeder, D. M., Blankenship, D. D. & Young, D. A. Evidence for a water system transition beneath Thwaites Glacier, West Antarctica. Proc. Natl Acad. Sci. USA 110 , 12225–12228 (2013).

Brunt, K. M., Fricker, H. A. & Padman, L. Analysis of ice plains of the Filchner–Ronne Ice Shelf, Antarctica, using ICESat laser altimetry. J. Glaciol. 57 , 965–975 (2011).

Begeman, C. B. et al. Ocean stratification and low melt rates at the ross ice shelf grounding zone. J. Geophys. Res. Oceans 123 , 7438–7452 (2018).

Milillo, P. et al. On the short-term grounding zone dynamics of Pine Island Glacier, West Antarctica, observed with cosmo-skymed interferometric data. Geophys. Res. Lett. 44 , 10–436 (2017).

Seroussi, H. et al. ISMIP6 Antarctica: a multi-model ensemble of the Antarctic ice sheet evolution over the 21st century. Cryosphere 14 , 3033–3070 (2020).

McPhee, M. G. Turbulent heat flux in the upper ocean under sea ice. J. Geophys. Res. Oceans 97 , 5365–5379 (1992).

Bradley, A. T., Rosie Williams, C., Jenkins, A. & Arthern, R. Asymptotic analysis of subglacial plumes in stratified environments. Proc. R. Soc. London 478 , 20210846 (2022).

Jenkins, A., Nicholls, K. W. & Corr, H. F. Observation and parameterization of ablation at the base of Ronne Ice Shelf, Antarctica. J. Phys. Oceanogr. 40 , 2298–2312 (2010).

Holland, D. M. & Jenkins, A. Modeling thermodynamic ice–ocean interactions at the base of an ice shelf. J. Phys. Oceanogr. 29 , 1787–1800 (1999).

Lazeroms, W. M., Jenkins, A., Rienstra, S. W. & Van De Wal, R. S. An analytical derivation of ice-shelf basal melt based on the dynamics of meltwater plumes. J. Phys. Oceanogr. 49 , 917–939 (2019).

Hewitt, I. J. Subglacial plumes. Annu. Rev. Fluid Mech. 52 , 145–169 (2020).

Kimura, S., Nicholls, K. W. & Venables, E. Estimation of ice shelf melt rate in the presence of a thermohaline staircase. J. Phys. Oceanogr. 45 , 133–148 (2015).

Patankar, S. V. Numerical Heat Transfer and Fluid Flow (CRC Press, 1980).

Lu, P., Li, Z., Cheng, B. & Leppäranta, M. A parameterization of the ice–ocean drag coefficient. J. Geophys. Res. Oceans 116 , C07019 (2011).

Ezhova, E., Cenedese, C. & Brandt, L. Dynamics of three-dimensional turbulent wall plumes and implications for estimates of submarine glacier melting. J. Phys. Oceanogr. 48 , 1941–1950 (2018).

Joughin, I. et al. Basal conditions for Pine Island and Thwaites glaciers, West Antarctica, determined using satellite and airborne data. J. Glaciol. 55 , 245–257 (2009).

Pattyn, F. Antarctic subglacial conditions inferred from a hybrid ice sheet/ice stream model. Earth Planet. Sci. Lett. 295 , 451–461 (2010).

Hewitt, I. J. Modelling distributed and channelized subglacial drainage: the spacing of channels. J. Glaciol. 57 , 302–314 (2011).

Carter, S. & Fricker, H. The supply of subglacial meltwater to the grounding line of the Siple Coast, West Antarctica. Ann. Glaciol. 53 , 267–280 (2012).

Bradley, A.T. & Hewitt, I.J. ‘Tipping-point in ice-sheet grounding-zone melting due to ocean water intrusion’ by Bradley and Hewitt. Zenodo https://doi.org/10.5281/zenodo.10895498 (2024).

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Acknowledgements

A.T.B. is supported by the Natural Environmental Research Programme (NERC) grant NE/S010475/1 and the PROTECT project, which received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 869304.

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A.T.B. conceived of the study and performed the analysis. A.T.B. and I.J.H. contributed to scientific discussion, interpretation of the results and wrote the manuscript.

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Bradley, A.T., Hewitt, I.J. Tipping point in ice-sheet grounding-zone melting due to ocean water intrusion. Nat. Geosci. (2024). https://doi.org/10.1038/s41561-024-01465-7

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Received : 11 May 2023

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Published : 25 June 2024

DOI : https://doi.org/10.1038/s41561-024-01465-7

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Identifying the coupling coordination relationship between urbanization and ecosystem services supply–demand and its driving forces: case study in shaanxi province, china.

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Liu, J.; Wang, H.; Hui, L.; Tang, B.; Zhang, L.; Jiao, L. Identifying the Coupling Coordination Relationship between Urbanization and Ecosystem Services Supply–Demand and Its Driving Forces: Case Study in Shaanxi Province, China. Remote Sens. 2024 , 16 , 2383. https://doi.org/10.3390/rs16132383

Liu J, Wang H, Hui L, Tang B, Zhang L, Jiao L. Identifying the Coupling Coordination Relationship between Urbanization and Ecosystem Services Supply–Demand and Its Driving Forces: Case Study in Shaanxi Province, China. Remote Sensing . 2024; 16(13):2383. https://doi.org/10.3390/rs16132383

Liu, Jiamin, Hao Wang, Le Hui, Butian Tang, Liwei Zhang, and Lei Jiao. 2024. "Identifying the Coupling Coordination Relationship between Urbanization and Ecosystem Services Supply–Demand and Its Driving Forces: Case Study in Shaanxi Province, China" Remote Sensing 16, no. 13: 2383. https://doi.org/10.3390/rs16132383

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