Question

A hollow sphere of external and internal diameters of $8cm$ and $4cm$ respectively is melted and made into another solid in the shape of a right circular cone of base diameter $8cm$. Find the height of the cone.

Answer 1

Let $R$ and $r$ be the external and internal radii of the hollow sphere.
Let $h$ and $r_1$ be the height and the radius of the cone to be made.
Hollow sphere
External:
$2R=8cm$
$\Rightarrow$ $R=4cm$
Internal
$2r=4cm$
$\Rightarrow$ $r=2cm$
Cone
$2r_1=8$
$\Rightarrow$ $r_1=4$
When the hollow sphere is melted and made into a solid cone, we have
Volume of the cone $=$ Volume of the hollow sphere
$\frac{1}{3}\pi r_1^{2}h=\frac{4}{3}\pi [R^3-r^3]$
$\frac{1}{3}\times \pi \times 4^2\times h=\frac{4}{3}\times \pi \times (4^3-2^3)$
$h=\frac{64-18}{4}=14$
Hence, the height of the cone $h=14cm$