Calculus help. Find the work required to empty a cone shaped tank.

Question

A tank in the shape of an inverted right circular cone (the ice cream goes on top) has height 7 meters and radius 3 meters. It is filled with 3 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 1050 kg/m^3.

Answer 1

The force due to gravity is $F=mg$.

The work required to lift a body through a height $h$ is $W=Fh=mgh$.

An infinitesimal slice of chocolate $\Delta y$ at height $y$ has a radius of $\frac{3}{7}y$, a volume of $\pi \frac{9}{49} y^2 \Delta y$ and mass of $\pi \frac{9}{49} y^2 \Delta y \rho$ and needs to be raised a height of $7-y$ which requires work of $\pi \frac{9}{49} y^2 \Delta y \rho g(7-y)$. To empty the tank requires:

$$\begin{align} W&=\int_0^3 \pi \frac{9}{49} \rho g (7y^2-y^3)dy\\ &=\pi \frac{9}{49} \rho g \left(7\frac{3^3}{3}-\frac{3^4}{4}\right)\\ &=\pi \frac{9}{49}\times 1050 \times 9.8 \times\frac{171}{4}\\ &\approx 254\text{kJ}\\ \end{align}$$