Question

The radii of internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respctively. It is melted and recast into a solid cylinder of diameter 14 cm. Find the height of the cylinder.

Answer 1

We have,

The internal base radius of the spherical shell, $r_1=3$ cm,

The external base radius of the spherical shell, $r_2=5$ cm and

The base radius of solid cylinder, $r=\frac{14}{2}=7$ cm

Let the height of the cylinder be h.

As,Volume of solid cylinder = Volume of spherical shell

$\Rightarrow \pi r^{2}h=\frac{4}{3}\pi r_2^3-\frac{4}{3}\pi r_1^3\Rightarrow \pi r^{2}h=\frac{4}{3}\pi (r_2^3-r_1^3)\Rightarrow r^{2}h=\frac{4}{3}(r_2^3-r_1^3)\Rightarrow 49\times h=\frac{4}{3}(125-27)\Rightarrow h=\frac{4}{3}\times \frac{98}{49}\therefore h=\frac{8}{3}$

So, the height of the cylinder is $\frac{8}{3}$