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At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool.

At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool. (b) Suppose that instead floor of the pool is 1 meter deep at one corner, 3 meters deep at the opposite corner, that the depth varies linearly on the line l between these corners, and that the pool's depth is constant on any line segment perpendicular to l (so the pool's floor is rotated and flattened from part (a)). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool. Hint: The bottom of the pool forms a portion of a plane that has equation z = Ax+By+C for some A, B, C. There are many ways to find this equation; for one possibility, note that depth is constant along lines of the form y = -2x + c. (c) Suppose that now the pool is circular with a radius of 25 meters, and that the pool has depth 1 meter on the circle's circumference, 3 meters in the center, and varies linearly along any radius. Set up and evaluate a double integral using polar coordinates to find the total pressure of water exerted on the bottom of the pool.
Transcribed Image Text:At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool. (b) Suppose that instead floor of the pool is 1 meter deep at one corner, 3 meters deep at the opposite corner, that the depth varies linearly on the line l between these corners, and that the pool's depth is constant on any line segment perpendicular to l (so the pool's floor is rotated and flattened from part (a)). Set up and evaluate a double integral to find the total pressure of water exerted on the bottom of the pool. Hint: The bottom of the pool forms a portion of a plane that has equation z = Ax+By+C for some A, B, C. There are many ways to find this equation; for one possibility, note that depth is constant along lines of the form y = -2x + c. (c) Suppose that now the pool is circular with a radius of 25 meters, and that the pool has depth 1 meter on the circle's circumference, 3 meters in the center, and varies linearly along any radius. Set up and evaluate a double integral using polar coordinates to find the total pressure of water exerted on the bottom of the pool.
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