how much work is required

Question

A water tank has the shape of a circular cone with its point downward, a height of 10 meters and a base radius of 4 meters. How much work is required to empty the tank by pumping all of the water to the top? (Water's weight per volume is 9800 N/m³.)

Answer 1

Put the point at the x-y axis origin.

Using cylindrical slices:

Slice volume = πr²h =πx²dy

4/10 = x/y by similar triangles

V = 0.16πy²dy

Force = (9800)(0.16)πy²dy

Distance per slice = (10-y)

Work = ∑Force×Distance

Work = (9800)(0.16)π ∫¹⁰₀(10-y)y²dy = (9800)(1.6)π[10y³/3-y⁴/4]¹⁰₀ = 3920000π/3