Question

The volume of two spheres are in the ratio 64 : 27. Find the difference of their surface areas if the sum of their radii is 7 cm.

Answer 1

Let the radii of the spheres be R cm and r cm.
Then we have:
$$\begin{gathered} \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3}=\frac{64}{27} \\ \Rightarrow {\left(\frac{R}{r}\right)}^3=\frac{64}{27} \\ \Rightarrow \frac{R}{r}=\frac{4}{3} \\ \mathrm{Now},r+R=7 \\ \mathrm{Then}\frac{7-r}{r}=\frac{4}{3} \\ \Rightarrow 21-3r=4r \\ \Rightarrow 7r=21 \\ \mathrm{i}.\mathrm{e}.,r=3\mathrm{cm}\mathrm{and}R=4\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{Difference}\mathrm{of}\mathrm{their}\mathrm{surface}\mathrm{areas}=4\mathrm{\pi }{R}^2-4\mathrm{\pi }{r}^2 \\ =4\times \frac{22}{7}\times \left(4^2-3^2\right) \\ =\frac{88}{7}\times 7 \\ =88{\mathrm{cm}}^2 \end{gathered}$$