Question

A solid cylinder has a TSA of $462c{m}^2$. Its CSA is one-third of its TSA. Find the volume of the cylinder.

Answer 1

CSA of cylinder $=\frac{1}{3}$ TSA of the cylinder $\therefore 2\pi rh=\frac{1}{3}\times 2\pi r(h+r)=\frac{1}{3}\times 462c{m}^2$
$\therefore 2\pi rh=154c{m}^2$
TSA of cylinder $=2\pi r(h+r)\therefore 462c{m}^2=2\pi rh+2\pi r^2=154c{m}^2+2\pi r^2$
$462c{m}^2-154c{m}^2=2\pi r^2\Rightarrow 308c{m}^2=2\times \frac{22}{7}\times r^2\Rightarrow \frac{308\times 7}{2\times 22}=r^2$
$r^2=49\therefore r=7$cm
CSA of cylinder $=154c{m}^2$ $\therefore 2\pi rh=154c{m}^2$
$2\times \frac{22}{7}\times 7cm\times h=154c{m}^2$ $\therefore h=\frac{154}{44}=\frac{7}{2}$cm
Volume of the cylinder $=\pi r^{2}h=\frac{22}{7}\times 7^2\times \frac{7}{2}c{m}^3=\frac{22}{7}\times 49\times \frac{7}{2}c{m}^3=539c{m}^3$.