Question

A water trough has a trapezoidal cross section with a height of 1 m and horizontal sides of length 1/2 m, and 1 m. Assume the length of the trough is 10 m.

10 m 1 m 0.5 m

(a) How much work is required to pump 0.001m thick layer of water at a depth of 0.25m to the top of the trough? Show all work.

(b) How much work is required to pump all the water out of the trough (to the level of the top of the trough) when it is full? Show all work.

(c) If the length of the trough is doubled to 20m, is the work required to empty the trough doubled? Why or why not? Fully explain/support your reasoning using complete sentences and / or calculus.

(d) If the trough is half full, is the work required to empty the trough half of the work required to empty the trough when it is full? Why or why not? Fully explain / support your reasoning using complete sentences and / or calculus.

Answer 1

Given that A cattle trough has a height of 1m and horizontal sides of width 1/2 m and 1 m place the centre of the base of the tank of the origin as shown in the below figure. Let y be the height of a rectangular layer of water above the described tank. Now, the distance from the y - axis is the line which connects the two points (0.25, 0) and (0.5, 1) \r\n y - 0 = [1 - 0/0.5 - 0.25] [x - 0.25] y = 1/0.25 (x - 0.25) x - 0.25 = 0.25 y x = 1/4 (y + 1) so, for any y elementof [0, 1], the width of the tank is given by, w(y) = 2x = 2[1/4 (y + 1)] = 1/2 (y + 1) Thus the rectangular layer of water has an area of A = 1.w = 10.1/2 (y + 1) = 5 (y + 1)