Question

The ratio between the volume of two spheres is 8 : 27. What is the ratio between their surface areas?

Options

• 2 : 3

• 4 : 5

• 5 : 6

• 4 : 9

Answer 1

4 : 9
Let the radii of the spheres be R and r, respectively.
Then, ratio of their volumes ${\displaystyle =\frac{\frac{4}{3}\pi {\text{R}}^3}{\frac{4}{3}\pi {\text{r}}^3}}$

Therefore,

${\displaystyle \frac{\frac{4}{3}\pi {\text{R}}^3}{\frac{4}{3}\pi {\text{r}}^3}=\frac{8}{27}}$

${\displaystyle \Rightarrow \frac{R^3}{{\text{r}}^3}=\frac{8}{27}}$

${\displaystyle \Rightarrow {\left(\frac{\text{R}}{\text{r}}\right)}^3={\left(\frac{2}{3}\right)}^3}$

${\displaystyle \Rightarrow \frac{R}{r}=\frac{2}{3}}$

Hence, the ratio between their surfcae areas ${\displaystyle =\frac{4\pi {\text{R}}^2}{4\pi {\text{r}}^2}}$

${\displaystyle =\frac{{\text{R}}^2}{{\text{r}}^2}}$

${\displaystyle ={\left(\frac{2}{3}\right)}^2}$

${\displaystyle =\frac{4}{9}}$

= 4 : 9