Question

The ratio between the volume of two spheres is 8 : 27. What is the ratio between their surface areas?
(a) 2 : 3
(b) 4 : 5
(c) 5 : 6
(d) 4 : 9

Answer 1

(d) 4 : 9
Let the radii of the spheres be R and r, respectively.
Then, ratio of their volumes$=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}$
Therefore,
$$\begin{gathered} \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}=\frac{8}{27} \\ \Rightarrow \frac{R^3}{r^3}=\frac{8}{27} \\ \Rightarrow {\left(\frac{R}{r}\right)}^3={\left(\frac{2}{3}\right)}^3 \\ \Rightarrow \frac{R}{r}=\frac{2}{3} \end{gathered}$$
Hence, the ratio between their surface areas$=\frac{4\mathrm{\pi }{R}^2}{4\mathrm{\pi }{r}^2}$
$$\begin{gathered} =\frac{R^2}{r^2} \\ ={\left(\frac{R}{r}\right)}^2 \\ ={\left(\frac{2}{3}\right)}^2 \\ =\frac{4}{9} \\ =4:9 \end{gathered}$$