A closed cylinder has volume \(500\pi\) cm3 and height \(20\) cm. (a) Find the radius of the cylinder. [2] (b) Find the curved surface area of the cylinder ...

(a) The volume of a cylinder is \(V = \pi r^{2} h\). Substituting the known values gives an equation in \(r\):

\(\pi r^{2} \times 20 = 500\pi\). Dividing both sides by \(\pi\) and then by \(20\) gives \(r^{2} = \dfrac{500}{20} = 25\) [M1].

\(r = \sqrt{25} = 5\) cm [A1] cao. The negative root is rejected because a radius is a length.

(b) The curved surface of a cylinder unrolls into a rectangle whose width is the circumference \(2\pi r\) and whose height is \(h\), so its area is \(2\pi r h\).

\(2\pi \times 5 \times 20\) [M1] \(= 200\pi\) cm\(^{2}\) [A1] cao.

Cancelling \(\pi\) at the start of part (a) is what makes the arithmetic exact without a calculator. Note that the question asks only for the curved surface; the total surface area of this closed cylinder would additionally include two circular ends of \(25\pi\) cm\(^{2}\) each.