Question:

A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is $\displaystyle 192\pi$$\displaystyle cm^2$. The cylinder has a radius of $\displaystyle r$$\displaystyle cm$ and a height of $\displaystyle h$$\displaystyle cm$.

(i) Express h in terms of r and show that the volume, $\displaystyle V$$\displaystyle cm^3$, of the cylinder is given by:

$\displaystyle V = \frac{1}{2}\pi(192r - r^3)$

Given that $\displaystyle r$ can vary,

(ii) Find the value of $\displaystyle r$ for which $\displaystyle V$ has a stationary value,

(iii) find this stationary value and determine whether it is a maximum or a minimum.

(i) Can someone please show me how to get $\displaystyle V = \frac{1}{2}\pi(192r - r^3)$?