A tank in the shape of an inverted right circular cone has h

A tank in the shape of an inverted right circular cone has height 10 meters and radius 6 meters. It is filled with 8 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. Note: the density of hot chocolate is =1490kg/m^(3) and we are using weight-density.

Solution

The radius of the liquid surface at any height y is r_y = (2 / 7) * y

The differential mass in a differential thickness of liquid at height

d m(y) = p * pi * r² * dy = p * pi * (4/49) * y² * dy

The distance to the rim of the cone is (7 - y)

The differential work to lift the differential mass to the rim of the cone is

dW = dm * g * h = p * pi * (4 / 49) * y² * g * (7 - y) * dy

dW = p * pi * (4 / 49) * g * (7 y² - y³)*dy

W = integral of dW from 0 to 5

W = p * pi * (4 / 49) * g * [ 7*y³/3 - y⁴/4 ] evaluated between 0 and 5

W = p * pi * (4/49) * g * [ 10*9 - 81/4 ]

W = 1070 * pi * ( 4 / 49 ) * 9.8 * ( 135.42 )

= 3.64 * 10⁵ J