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Chapter 6 | Applications of Integration
Example 6.28 Chapter Opener: Finding Hydrostatic Force
We now return our attention to the Hoover Dam, mentioned at the beginning of this chapter. The actual dam is arched, rather than flat, but we are going to make some simplifying assumptions to help us with the calculations. Assume the face of the Hoover Dam is shaped like an isosceles trapezoid with lower base 750 ft, upper base 1250 ft, and height 750 ft (see the following figure).
When the reservoir is full, Lake Mead’s maximum depth is about 530 ft, and the surface of the lake is about 10 ft below the top of the dam (see the following figure).
Figure 6.59 A simplified model of the Hoover Dam with assumed dimensions.
a. Find the force on the face of the dam when the reservoir is full. b. The southwest United States has been experiencing a drought, and the surface of Lake Mead is about 125 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances? Solution a. We begin by establishing a frame of reference. As usual, we choose to orient the x -axis vertically, with the downward direction being positive. This time, however, we are going to let x =0 represent the top of the dam, rather than the surface of the water. When the reservoir is full, the surface of the water is 10 ft below the top of the dam, so s ( x ) = x −10 (see the following figure).
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