Pumping water out of rectangular tank

Question

3ft by 4ft by 10ft high

tank filled to half way level

wt of water 62.26lbs/cu ft

How much work is done to pump all water out top of tank

Answer 1

Draw a sketch:

We can set up our coordinate plane at the top of the tank with positive going down to make it easier. That makes the top of the water at y = 5 and the bottom of the water at y = 10.

To lift that little red section of water out of the tank, you have to apply a force equal to its weight over the distance of "y". The weight of that red section is its volume times the density (62.26 lbs/ft³). Its volume is length • width • depth = (3)(4)(dy) or 12dy. So its weight is (62.26)(12dy) = 747.12dy. The work necessary to lift that weight is (y)(747.12dy). Then you must add all the little red sections up from 5 to 10 by integrating so the total work is:

₅∫¹⁰ 747.12y dy

The antiderivative of 747.12y is 373.56y² so we need to evaluate this between 5 and 10:

373.56(10)² - 373.56(5)² = 28017 ft lbs of work