Question

Tick (✓) the correct answer:
Two cubes have their volumes in the ratio 1 : 27. The ratio of their surface areas is
(a) 1 : 3
(b) 1 : 9
(c) 1 : 27
(d) none of these

Answer 1

(b) 1: 9

$$\begin{gathered} \frac{\mathrm{Volume}1}{\mathrm{Volume}2}=\frac{1}{27}=\frac{a^3}{b^3} \\ \Rightarrow a=\frac{b}{\sqrt[3]{27}}=\frac{b}{3}\mathrm{or}b=3a\mathrm{or}\frac{b}{a}=3 \end{gathered}$$
$$\begin{gathered} \mathrm{Now},\frac{\mathrm{surface}\mathrm{area}1}{\mathrm{surface}\mathrm{area}2}=\frac{6{\mathrm{a}}^2}{6{\mathrm{b}}^2}=\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}=\frac{(\mathrm{b}/3{)}^2}{{\mathrm{b}}^2}=\frac{1}{9} \\ \therefore \mathrm{Ratio}\mathrm{of}\mathrm{the}\mathrm{surface}\mathrm{areas}=1:9 \end{gathered}$$