A tank filled with oil is in the shape of a downward-pointing cone with its vert

Question

A tank filled with oil is in the shape of a downward-pointing cone with its vertical axis perpendicular to ground level. (See a graph of the tank here.) In this exercise we will assume that the height of the tank is h = 12 feet, the circular top of the tank has radius r = 6 feet, and that the oil inside the tank weighs 57 pounds per cubic foot. How much work (W) would it take to pump oil from the tank to the level at the top of the tank if the tank were completely full? W = ft-lb (Do NOT include units in the box above, or commas in the definitions of large numbers. That is, write 104500 instead of 104, 500.)

Explanation / Answer

Measuring height from the apex of the cone (at the bottom) the centre of gravity of the oil contained in the conical tank is at 12 x 0.75 = 9 feet.
So the potential energy gain in lifting all the oil to the top is ( 12 -9) x weight of oil in tank.

the mass of oil in the tank = pi x r^2 x h/3 x density = pi x 36 x 12 x 57/3 = 25786.19250 lb.

So work (energy) required = 3 x 25786.19250= 77358.5775. ft-lbf.

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