Question

The volumes of the two spheres are in the ratio 64 : 27 and the sum of their radii is 7 cm. The difference of their total surface areas is
(a) 38 cm²
(b) 58 cm²
(c) 78 cm²
(d) 88 cm²

Answer 1

(d) 88 cm²

Suppose that the radii of the spheres are r cm and (7 − r) cm. Then we have:
$$\begin{gathered} \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{\left(7-\mathrm{r}\right)}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}=\frac{64}{27} \\ \Rightarrow \frac{(7-r)}{r}=\frac{4}{3} \\ \Rightarrow 21-3r=4r \\ \Rightarrow 21=7r \\ \Rightarrow r=3\mathrm{cm} \end{gathered}$$

Now, the radii of the two spheres are 3 cm and 4 cm.

$$\begin{gathered} \therefore \mathrm{Required}\mathrm{difference}=4\times \frac{22}{7}\times 4^2-4\times \frac{22}{7}\times 3^2 \\ =4\times \frac{22}{7}\times \left(16-9\right) \\ =4\times \frac{22}{7}\times 7 \\ =88{\mathrm{cm}}^2 \end{gathered}$$