A closed cylinder has total surface area $24\pi$ cm². What is its largest possible volume, in cm³?

Explanation

1. 01

   $2\pi r^{2} + 2\pi rh = 24\pi$ gives $h = \frac{12 - r^{2}}{r}$, so $V = \pi r^{2}h = \pi(12r - r^{3})$.

2. 02

   $V' = \pi(12 - 3r^{2}) = 0$ at $r = 2$, and $V'' = -6\pi r < 0$: then $h = 4$ and $V = \pi \times 4 \times 4 = 16\pi$.