Ans: (a) $288\pi\text{ cm}^3$ (b) $180\pi\text{ cm}^2$
- Let $r\text{ cm}$ be the radius of the larger sphere.
Since the two spheres are similar, then the ratio of the volume of the larger sphere to that of the smaller sphere
$\begin{array}{cl}
= & \left( \dfrac{r}{\frac{1}{2}r} \right)^3 \\
= & \dfrac{8}{1} \\
= & 8 : 1
\end{array}$Hence, the volume of the larger sphere
$\begin{array}{cl}
= & 324\pi \times \dfrac{8}{8 + 1} \\
= & 288\pi \text{ cm}^3
\end{array}$ - By the result of (a), we have
$\begin{array}{rcl}
\dfrac{4}{3} \pi r^3 & = & 288\pi \\
r^3 & = & 216 \\
r & = & 6
\end{array}$Therefore, the radii of the larger sphere and smaller sphere are $6\text{ cm}$ and $3\text{ cm}$ respectively.
Hence, the sum of the surface areas of the two spheres
$\begin{array}{cl}
= & 4 \pi (6)^2 + 4 \pi (3)^2 \\
= & 180\pi \text{ cm}^2
\end{array}$
