EXAMPLE 1 A dam has the shape of a trapezoid shown in the top figure. The height is 30 m and the width is 60 m at the top and 30 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water level is 2 m from the top of the dam. SOLUTION We choose a vertical x-axis with origin at the surface of the water and directed downward as in the middle figure. The depth of the water is 28 m, so we divide the interval [0, 28] into subintervals of equal length with endpoints x; and we choose x E [x; - 1, x;]. The ith horizontal strip of the dam is approximated by a rectangle with height Ax and width wi, where, from similar triangles in the bottom figure, or a = %3D 24 - Xị 30 2 and so 2(15 + |× -x;* w; = 2(15 + a) = = If Aj is the area of the ith strip, then Aj x wjAx = - Xj If Ax is small, then the pressure Pj on the ith strip is almost constant and we can use this equation to write Pi x 1000gx;" The hydrostatic force F; acting on the ith strip is the product of the pressure and the area: Fj = PA; x 1000gx;"|| -X Adding these forces and taking the limit as n- o, we obtain the total hydrostatic force on the dam: F= lim 1000gx ( i = 1 28 1000gx dx 28 = 1000(9.8) dx 28ך = 9800 N (round to the nearest integer)
EXAMPLE 1 A dam has the shape of a trapezoid shown in the top figure. The height is 30 m and the width is 60 m at the top and 30 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water level is 2 m from the top of the dam. SOLUTION We choose a vertical x-axis with origin at the surface of the water and directed downward as in the middle figure. The depth of the water is 28 m, so we divide the interval [0, 28] into subintervals of equal length with endpoints x; and we choose x E [x; - 1, x;]. The ith horizontal strip of the dam is approximated by a rectangle with height Ax and width wi, where, from similar triangles in the bottom figure, or a = %3D 24 - Xị 30 2 and so 2(15 + |× -x;* w; = 2(15 + a) = = If Aj is the area of the ith strip, then Aj x wjAx = - Xj If Ax is small, then the pressure Pj on the ith strip is almost constant and we can use this equation to write Pi x 1000gx;" The hydrostatic force F; acting on the ith strip is the product of the pressure and the area: Fj = PA; x 1000gx;"|| -X Adding these forces and taking the limit as n- o, we obtain the total hydrostatic force on the dam: F= lim 1000gx ( i = 1 28 1000gx dx 28 = 1000(9.8) dx 28ך = 9800 N (round to the nearest integer)
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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![EXAMPLE 1
A dam has the shape of a trapezoid shown in the top figure. The height is 30 m and the width is 60 m at the top and 30 m at the bottom.
Find the force on the dam due to hydrostatic pressure if the water level is 2 m from the top of the dam.
SOLUTION We choose a vertical x-axis with origin at the surface of the water and directed downward as in the middle figure. The depth of the water is
28 m, so we divide the interval [0, 28] into subintervals of equal length with endpoints x; and we choose x E [X; - 1, x]. The ith horizontal strip of the
dam is approximated by a rectangle with height Ax and width wi, where, from similar triangles in the bottom figure,
a
15
Xị
or
a =
24
- Xj
30
2
2
and so
- 2(1
]-) - 54
wj = 2(15 + a) =
If Aj is the area of the ith strip, then
Aj z WiAx =
If Ax is small, then the pressure P; on the ith strip is almost constant and we can use this equation to write
Pi x 1000gx;"
The hydrostatic force F; acting on the ith strip is the product of the pressure and the area:
Fj = PA; x 1000gx;
Adding these forces and taking the limit as n- 00, we obtain the total hydrostatic force on the dam:
F= lim
n - co
1000gx; (
- X
x) dx
1000
28
= 1000(9.8)
xp
128
= 9800
N (round to the nearest integer)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0af9ba22-2332-42bc-9ff9-81afa8fe4a56%2Fcb29c2b0-c4a6-4177-b754-11ab91af0d67%2F1yo5mxk_processed.png&w=3840&q=75)
Transcribed Image Text:EXAMPLE 1
A dam has the shape of a trapezoid shown in the top figure. The height is 30 m and the width is 60 m at the top and 30 m at the bottom.
Find the force on the dam due to hydrostatic pressure if the water level is 2 m from the top of the dam.
SOLUTION We choose a vertical x-axis with origin at the surface of the water and directed downward as in the middle figure. The depth of the water is
28 m, so we divide the interval [0, 28] into subintervals of equal length with endpoints x; and we choose x E [X; - 1, x]. The ith horizontal strip of the
dam is approximated by a rectangle with height Ax and width wi, where, from similar triangles in the bottom figure,
a
15
Xị
or
a =
24
- Xj
30
2
2
and so
- 2(1
]-) - 54
wj = 2(15 + a) =
If Aj is the area of the ith strip, then
Aj z WiAx =
If Ax is small, then the pressure P; on the ith strip is almost constant and we can use this equation to write
Pi x 1000gx;"
The hydrostatic force F; acting on the ith strip is the product of the pressure and the area:
Fj = PA; x 1000gx;
Adding these forces and taking the limit as n- 00, we obtain the total hydrostatic force on the dam:
F= lim
n - co
1000gx; (
- X
x) dx
1000
28
= 1000(9.8)
xp
128
= 9800
N (round to the nearest integer)
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