Question

If the ratio of the total surface areas of two cubes is $9:16$, what is the ratio of their volumes ?

• A. $27:64$
• B. $3:4$
• C. $3:8$
• D. $729:216$

Answer 1

Answer: (A)

$27:64$

Let the total surface areas of two cubes be $S_1$ and $S_2$

We know, total surface area of a cube $S=6x^2$.

Then, $\frac{S_1}{S_2}=\frac{6x_1^2}{6x_2^2}=\frac{9}{16}$

$\Rightarrow {\left(\frac{x_1}{x_2}\right)}^2=\frac{9}{16}$

$\Rightarrow \frac{x_1}{x_2}=\sqrt{\frac{9}{16}}=\frac{3}{4}$.

Also, Volume of cube $V=x^3$.

$\therefore$ Ratio of volumes $=\frac{V_1}{V_2}=\frac{x_1^3}{x_2^3}={\left(\frac{x_1}{x_2}\right)}^3$

$={\left(\frac{3}{4}\right)}^3$ $=\frac{27}{64}$ $=27:64$.

Hence, option $A$ is correct.