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