1.55: A solid right circular cylinder has base radius cm and height cm, as shown in Figure 1.

A solid right circular cylinder has base radius $\displaystyle r$ cm and height $\displaystyle h$ cm, as shown in Figure 1.

The cylinder has a total surface area of $\displaystyle 322\pi\,\mathrm{cm}^2$ and a volume of $\displaystyle V\,\mathrm{cm}^3$.

(a) Show that

Given that $\displaystyle r$ can vary,

(b) use calculus to find, to 3 significant figures, the value of $\displaystyle r$ for which $\displaystyle V$ is a maximum, justifying that this value of $\displaystyle r$ gives a maximum value of $\displaystyle V$.

(c) Find, to 3 significant figures, the maximum value of $\displaystyle V$.