calculate work follow through description

Question

A tank in the shape of an inverted right circular cone has height 5 meters and radius 4 meters. It is filled with 2 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 𝛿=1080 kg/m3.δ=1080 kg/m3. Your answer must include the correct units.

Answer 1

The work equals the change in potential energy of each chocolate layer of elevation z, infinitesimal thickness dz, radius r(z) i.e., area a(z) = πr(z)² and hence infinitesimal weight df(z) = gδa(z)dz, If "inverted" means cone apex down at z = 0 < h = cone height then r(z) = r(h)h⁻¹z. The elevation change to get to the top (inverted base) is h–z so the total work to raise a depth 0 < c < h of chocolate is

w(c) = ∫₀^c(h – z)df(z)

= gδπ ∫₀^c(h – z)r(z)²dz

= gδπr(h)²h⁻² ∫₀^c(h – z)z²dz

= gδπr(h)²h⁻²(hc³/3 – c⁴/4)

= gδπr(h)²h²(1/3 – c/4h)(c/h)³

that for 0 < c < h increases monotonically from w(0) = 0 to w(h) = gδπr(h)²h²/12.