Question

The volumes of two cubes are in the ratio 1 : 27. The ratio of their surface area is
(a) 1 : 3
(b) 1 : 8
(c) 1 : 9
(d) 1 : 18

Answer 1

(c) 1 : 9
Let the edges of the two cubes be a and b.
Then, ratio of their volumes$=\frac{a^3}{b^3}$
Therefore,
$$\begin{gathered} \frac{a^3}{b^3}=\frac{1}{27} \\ \Rightarrow {\left(\frac{a}{b}\right)}^3={\left(\frac{1}{3}\right)}^3 \\ \Rightarrow \frac{a}{b}=\frac{1}{3} \end{gathered}$$
The ratio of their surface areas$=\frac{6a^2}{6b^2}$
Therefore,
$$\begin{gathered} \frac{6a^2}{6b^2}=\frac{a^2}{b^2} \\ =\frac{1}{9}\left[\text{Since}\frac{a}{b}=\frac{1}{3}\right] \\ =1:9 \end{gathered}$$
Hence, the ratio of their surface areas is 1:9.