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

Question

A lank filled with oil is in the shape of a downward-pointing cone with its vertical axis perpendicular to ground level. (Sec a graph of the tank here.) In this exercise we will assume that the height of the tank is h = 8 feet, the circular top of the tank has radius r = 4 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 = foot-pounds (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

The area of the top of the cone is (Pi 6^2) = 36Pi. The volume of a cone is one third the base area times the altitude. Thus the total volume of the complete cone is V = (1/3) (36 Pi) (15) = 565.49 cubic feet. When the oil is 5 feet deep the diameter of its top is (5/15)(12) = 4 feet. The initial volume of the oil is therefore (1/3) (Pi 2^2) (5) = 20.94 cubic feet. The volume needed to fill the tank is the difference of the total volume and starting volume: 565.49 - 20.94 = 544.55 cubic feet The time needed to fill the tank is therefore ( 544.55 cubic feet ) / ( 5 cubic feet / minute ) = 108.91 minutes Round that up to the nearest minute: 109 minutes = 1 hour 49 minutes So if the oil begins flowing at 1PM the tank will be full at 2:49PM Note that we could have saved a little work in the computation of the amount needed to fill the tank. Since the original height was one third the total height, and since the volume of a cone increases as the cube of the height, the amount needed is 565.49( 1 - (1/3)^3) = 544.55 cubic feet.

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