Question

If a sphere has the same surface area as the total surface area of a circular cone of height 40 cm and radius 30 cm, find the radius of the sphere.

Answer 1

Given: radius of the cone (r) = 30 cm and height of the cone (h) = 40 cm
Slant height of the cone (l) = $\sqrt{r^2+h^2}=\sqrt{{\left(30\right)}^2+{\left(40\right)}^2}$
$\Rightarrow \sqrt{900+1600}\Rightarrow \sqrt{2500}=50$ cm
Total surface area of the circular cone = $\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$ sq. units
$=\mathrm{\pi }\times 30\left(50+30\right)$ cm²
TSA of a sphere is same as TSA of a circular cone.
Therefore,
Total surface area of a sphere = Total surface area of circular cone
$4{\mathrm{\pi r}}^2=\mathrm{\pi }\times 30\times 80$
$\Rightarrow {\mathrm{r}}^2=\frac{\left(30\times 80\right)}{4}=600$
$\Rightarrow \mathrm{r}=10\sqrt{6}$ cm
Hence, radius of the sphere will be $10\sqrt{6}$cm.