Find the work required

Question

A tank in the shape of an inverted right circular cone has height 7 meters and radius 3 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is (delta)=1050 kg/m3. Your answer must include the correct units.

Answer 1

Work = Force×Distance

Force = Density×Volume, Density=ρ=1050

Put the vertex of the inverted cone at the origin of the x-y axis.

Using a small disk

Volume_disk = area×Δy = πx²Δy

By similar triangles, x/y = 3/7 --> x = (3/7)y

Volume_disk = π(9/49)y²Δy

Distance to move a small disk over the top = 7-y

Work = πρ(9/49)∫⁵₀(7-y)y²dy = πρ(9/49)∫⁵₀7y²-y³dy = πρ(9/49)[7y³/3-y⁴/4]⁵₀

≈ 82046.1 Joules