Chapter 6 | Applications of Integration
697
Figure 6.58 shows the trough and a more detailed view of one end.
Figure 6.58 (a) A water trough with a triangular cross-section. (b) Dimensions of one end of the water trough.
Select a frame of reference with the x -axis oriented vertically and the downward direction being positive. Select the top of the trough as the point corresponding to x =0 (step 1). The depth function, then, is s ( x ) = x . Using similar triangles, we see that w ( x ) =8−(8/3) x (step 2). Now, the weight density of water is 62.4 lb/ft 3 (step 3), so applying Equation 6.13 , we obtain F = ∫ a b ρw ( x ) s ( x ) dx = ∫ 0 3 62.4 ⎛ ⎝ 8− 8 3 x ⎞ ⎠ xdx =62.4 ∫ 0 3 ⎛ ⎝ 8 x − 8 3 x 2 ⎞ ⎠ dx =62.4 ⎡ ⎣ 4 x 2 − 8 9 x 3 ⎤ ⎦ | 0 3 =748.8. The water exerts a force of 748.8 lb on the end of the trough (step 4).
6.27 A water trough 12 m long has ends shaped like inverted isosceles triangles, with base 6 m and height 4 m. Find the force on one end of the trough if the trough is full of water.
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