A tank (shown below) is shaped like a circular cylinder measuring 20 feet high,

Question

A tank (shown below) is shaped like a circular cylinder measuring 20 feet high, with a radius of 5 feet. It is full of water, which weighs 62.4 lb/ft 5 ft 20 ft Suppose that there is an out flow pipe at the top of the tank, and water is pumped out of the tank through this outflow pipe. (a) Consider a slice of water that is y feet above the bottom of the tank and is Y feet thick (shown below). How much does this slice weigh? (b) Approximately how far is this slice displaced as it is pumped to the top of the tank? (c) Approximately how much work is done in pumping this slice to the top of the tank? (d) Now imagine adding up your answers to part (c) over all of the slices of water, and then taking a limit as the thickness of the slices goes to zero. Use this to set up an integral that calculates the total work in pumping all of the water to the top of the tank. (e) Calculate the integral from part (d). Give a numerical answer, including units. (f) How would your answer change if the tank were only half full? Set up an integral to calculate the work in this case; you do not need to compute the integral.

Explanation / Answer

(a)

weight of slice =62.4**52*y

weight of slice =1560y

(b)

distance slice displaced =20-y

(c)

work done in pumping the slice to top of tank =1560(20-y )y

(d)

total work =[0 to 20]1560(20-y )dy

(e)

total work =[0 to 20]1560(20y-(1/2)y2)

total work =1560(20*20-(1/2)202) -1560(20*0-(1/2)02)

total work =1560(400-200) -0

total work =312000 lb-ft

total work =980176.91 lb-ft

(f)

half full:

total work =[0 to 10]1560(20-y)dy

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