Question

The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder.

Answer 1

Let R₁ and R₂ be the internal and external radius of the hollow spherical shell respectively.

$R_1=\frac{6}{2}=3cmand{R}_2=\frac{10}{2}=5cm$

Volume of the hollow spherical shell

$=\frac{4}{3}\times \pi (R_1^3-R_2^3)$

$=\frac{4}{3}\times \pi (5^3-3^3)$

Suppose r and h be the radius and height of the cylinder respectively.

$r=\frac{14}{2}=7cm$

Volume of the cylinder $=\pi r^{2}h=\pi \times 7^2\times h$

If the hollow spherical shell is melted to form a solid cylinder, then

Volume of the cylinder = Volume of the hollow spherical shell

$\pi \times 7^2\times h=\frac{4}{3}\times \pi (5^3-3^3)$

$49h=\frac{4}{3}\times (125-27)cm$

$49h=\frac{4}{3}\times 98cm$

$h=\frac{4\times 98}{3\times 49}=\frac{8}{3}cm$

Thus, the height of the cylinder is 2.67 cm