Question

A tank in the shape of an inverted right circular cone has height 6 meters and radius 3 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

Answer 1

Answer:

Step-by-step explanation:

Given

Radius of cone $R=3\ m$

height of cone $H=6\ m$

Chocolate is filled up to a height of $h=2\ m$

Work done to lift a body to a height h when F force is applied is  given by

$W=Fh$

An infinitesimally slice of chocolate $\Delta y$ at a height of y has a radius

$r=\dfrac{R}{H}\cdot h$

Volume of slice is $dV=\pi r^2y$

$mass =\rho \times dV$

This mass need to be raised to a height of 6-y which requires a work of

$dW=\rho (\pi r^2ydy)g(6-y)$

To empty the tank

$W=\int_{0}^{3}\rho \cdot \pi \cdot g\cdot r^2\cdot (6-y)dy$

at $h=y$

$r=\dfrac{R}{H}\cdot y$

$r=\dfrac{3}{6}y$

$r=\dfrac{y}{2}$

$W=\int_{0}^{3}\rho \cdot \pi \cdot g\cdot \dfrac{y^2}{4}\cdot (6-y)dy$

$W=10^3\times \pi \times \dfrac{1}{4}\times (2y^3-\dfrac{1}{4}y^4)_0^3$

$W=259.80\ kJ$