Question

A solid cylinder has a total surface area of $231{\text{cm}}^2$. Its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder..

Answer 1

Let the radius of the cylinder be $r$ and the height be $h$.

The curved surface area of the cylinder is given by

$2\pi rh$

Total surface area of the solid cylinder is given as the sum of the curved surface area and the areas of the base and the top:

$2\pi r^2+2\pi rh$

Given that

$2\pi rh=\frac{2}{3}(2\pi rh+2\pi r^2)$

$\Rightarrow h=2r$

Also, given that

$2\pi r^2+2\pi rh=231$

i.e

$2\pi r^2+2\pi r.2r=231$

$6\pi r^2=231$

$\Rightarrow r=\sqrt{\frac{77}{2\pi }}$

Volume of the cylinder:

$=\pi r^{2}h$

$=\pi r^2.2r=2\pi r^3$

$=2\pi (\frac{77}{2\pi }{)}^{3/2}$

$=77\sqrt{\frac{77}{2\pi }}$

Hence, volume of the cylinder $=77\sqrt{\frac{77}{2\pi }}c{m}^3$