Question

Two cubes have their volumes in the ratio $1:27$ . Find the ratio of their surface areas.

Answer 1

Hint-
In this particular type of question we need to proceed by expressing the sides of the cubes as a and b and finding the ratio between them. Then we need to use it in finding the ratio of surface area of both the cubes.

Complete step-by-step answer:

Let the sides of the cube be a and b.
Then,
We know that volume of a cube = $sid{e}^3$
Therefore ratio of the volumes = ${\displaystyle \frac{a^3}{b^3}}$
But ratio = $1:27$
$$\begin{gathered} \Rightarrow {\displaystyle \frac{a^3}{b^3}}={\displaystyle \frac{1}{27}} \\ \Rightarrow {\left({\displaystyle \frac{a}{b}}\right)}^3={\left({\displaystyle \frac{1}{3}}\right)}^3 \\ \Rightarrow {\displaystyle \frac{a}{b}}={\displaystyle \frac{1}{3}} \end{gathered}$$
Therefore,
Ratio of the surface areas = ${\displaystyle \frac{6a^2}{6b^2}}$
(since surface area of the cube = $6\times sid{e}^2$ )
$$\begin{gathered} ={\displaystyle \frac{a^2}{b^2}} \\ ={\left({\displaystyle \frac{a}{b}}\right)}^2 \\ ={\left({\displaystyle \frac{1}{3}}\right)}^2\left(\text{putting value of }{\displaystyle \frac{a}{b}}\text{ from above}\right) \\ ={\displaystyle \frac{1}{9}} \\ =1:9 \end{gathered}$$
Therefore ratio of the surface areas =$1:9$.

Note- Always recall the formula of volume and surface area of three dimensional figures like cube. Note that a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.