Question

If the ratio of the volumes of two spheres is 8 : 27, then the ratio of their surface areas is __________.

Answer 1

Let r and R be the radii of the two spheres.

Suppose V₁ be the volume and S₁ be the surface area of the sphere of radius r & V₂ be the volume and S₂ be the surface area of the sphere of radius R.

It is given that the ratio of the volumes of two spheres is 8 : 27.

$\therefore \frac{V_1}{V_2}=\frac{8}{27}$

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

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

$\Rightarrow \frac{r}{R}=\frac{2}{3}$ .....(1)

Now,

$\frac{S_1}{S_2}=\frac{4\mathrm{\pi }{r}^2}{4\mathrm{\pi }{R}^2}$

$\Rightarrow \frac{S_1}{S_2}={\left(\frac{r}{R}\right)}^2$

$\Rightarrow \frac{S_1}{S_2}={\left(\frac{2}{3}\right)}^2$ [Using (1)]

$\Rightarrow \frac{S_1}{S_2}=\frac{4}{9}$

⇒ S₁ : S₂ = 4 : 9

Thus, the ratio of the surface areas of the two spheres is 4 : 9.

If the ratio of the volumes of two spheres is 8 : 27, then the ratio of their surface areas is ___4 : 9___.