1.48: Past-paper question 6

Figure 1 shows a solid right circular cylinder with radius $\displaystyle r$ cm and height $\displaystyle h$ cm

The total surface area of the cylinder is $\displaystyle 700\pi$$\displaystyle \mathrm{cm}^{2}$

The volume of the cylinder is $\displaystyle V$$\displaystyle \mathrm{cm}^{3}$

(a) Show that $\displaystyle V = \pi r(350 - r^2)$

Given that $\displaystyle r$ can vary and using calculus,

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

(c) Find, to 3 significant figures, the height $\displaystyle h$ cm for which $\displaystyle V$ is a maximum.