Question

Two cubes have volumes in the ratio 1 : 64. The ratio of the area of a face of first cube to that of the other is
a) 1 : 4
b) 1 : 8
c) 1 : 16
d) 1 : 32

Answer 1

Let $a$ and $b$ be the edges of the two cubes respectively.
Then, according to the question,
$a^3:b^3=1:64$ [$\because$ volume of cube $=(edge{)}^3$]
$\Rightarrow \frac{a^3}{b^3}$ $=\frac{1}{64}$
$\Rightarrow {\left(\frac{a}{b}\right)}^3={\left(\frac{1}{4}\right)}^3$
$\Rightarrow \frac{a}{b}=\frac{1}{4}$ [taking cube roots on both sides]
Now, ratio of area of the faces = ${\left(\frac{a}{b}\right)}^2$
$\Rightarrow (\frac{a}{b}{)}^2={\left(\frac{1}{4}\right)}^2$
$\Rightarrow \frac{a^2}{b^2}=\frac{1}{16}$

$\therefore a^2:b^2=1:16$