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)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.3: Hyperbolas

Problem 36E

See similar textbooks