The volumes of two cubes are in the ratio 1 : 27.
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

Solution:

(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,

$\frac{a^{3}}{b^{3}}=\frac{1}{27}$

$\Rightarrow\left(\frac{a}{b}\right)^{3}=\left(\frac{1}{3}\right)^{3}$

$\Rightarrow \frac{a}{b}=\frac{1}{3}$

The ratio of their surface areas $=\frac{6 a^{2}}{6 b^{2}}$

Therefore,

$\frac{6 a^{2}}{6 b^{2}}=\frac{a^{2}}{b^{2}}$

$=\frac{1}{9} \quad\left[\right.$ Since $\left.\frac{a}{b}=\frac{1}{3}\right]$

$=1: 9$

Hence, the ratio of their surface areas is 1:9.

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