Question

Two cubes have volumes in the ratio $1:27$. The ratio of the area of the face of one cube to that of the other is __________.

• A. $1:3$
• B. $1:6$
• C. $1:9$
• D. $1:18$

Answer 1

The correct option is C $1:9$

Given, two cubes have volume in ratio $1:27$.

Let $x_1$ and $x_2$ be edges of the cubes.

Then, volume of a cube $V=x^3$, where $x$ is the edge.

$\therefore \frac{V_1}{V_2}=\frac{1}{27}$

$\Rightarrow \frac{(x_1{)}^3}{(x_2{)}^3}=\frac{1}{27}$

$\Rightarrow {\left(\frac{x_1}{x_2}\right)}^3=\frac{1}{27}$
$\Rightarrow \frac{x_1}{x_2}=\sqrt[3]{3\frac{1}{27}}=\frac{1}{3}$.

We know, area of a cube $A=6x^2$.
Then, ratio of area $=\frac{A_1}{A_2}=\frac{6x_1^2}{6x_2^2}={\left(\frac{x_1}{x_2}\right)}^2$
$={\frac{1}{3}}^2=\frac{1}{9}=1:9$.

Hence, option $C$ is correct.