A rectangular tank that is 6 feet long, 9 feet wide and 6 feet deep is filled with a heavy liquid that weighs 120 pounds per cubic foot.

Question

A rectangular tank that is 6 feet long, 9 feet wide and 6 feet deep is filled with a heavy liquid that weighs 120 pounds per cubic foot. In each part below, assume that the tank is initially full. Your answers must include the correct units. (You may enter lbf or lb*ft for ft-lb.)

(a) How much work is done pumping all of the liquid out over the top of the tank? _______

(b) How much work is done pumping all of the liquid out of a spout 6 feet above the top of the tank? _______

(c) How much work is done pumping two-thirds of the liquid out over the top of the tank? _________

(d) How much work is done pumping two-thirds of the liquid out of a spout 6 feet above the top of the tank? _______

Answer 1

For each of these, think of sectioning each depth level of the pool into small layers that are rectangular. This means that the volume of that layer will be 6*9*dx or 54 dx cu. feet.

Now lets convert that to pounds: each layer is 120*54 dx = 65340 dx pounds.

Next, let's think about the work that's necessary to pump that layer out of the tank. In problems (a) and (c), this is simply the depth of the tank: If we think of the bottom of the tank being x=6, then the depth is x feet. In problems (b) and (d) this is the depth of the tank plus the additional 6 feet over the tank: depth is x+6.

Now simply multiply the weight of each layer by the depth and g=32.174 ft/sec² (represents how much effort it takes to lift the substance here on earth; Force = mass*acceleration) and integrate; for (a) and (b) the bounds are 0 to 6; for (c) and (d) the bounds are 0 to 4 (2/3 of the tank).

a) ∫₀⁶ 2102249x dx = 37840485 ft. lb.

b) ∫₀⁶ 2102249(x+6) dx = 113521455 ft. lb.

c) ∫₀⁴ 2102249x dx = 16817993 ft. lb.

d) ∫₀⁴ 2102249(x+6) dx = 67271973 ft. lb.