Question

Two circular cylinders of equal volume have their heights in the ratio 1 : 2. The ratio of their radii is
(a) $1:\sqrt{2}$
(b) $\sqrt{2}:1$
(c) 1 : 2
(d) 1 : 4

Answer 1

(b) $\sqrt{2}:1$

Suppose that the heights of two cylinders are h and 2h whose radii are r and R, respectively.
Since the volumes of the cylinders are equal, we have:

$$\begin{gathered} \mathrm{\pi }{r}^{\mathit{2}}h=\mathrm{\pi }{R}^2\times 2h \\ \Rightarrow \frac{r^{\mathit{2}}}{R^{\mathit{2}}}=\frac{2}{1} \\ \Rightarrow \left(\frac{r}{R}\right)=\sqrt{\frac{2}{1}} \\ \Rightarrow r\mathit{:}R\mathit{}=\sqrt{2}:1 \end{gathered}$$