Question

The total surface area of a solid cylinder is $231c{m}^2$ and its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder.

Answer 1

Given: Total surface area (TSA) of cylinder $=231c{m}^2$

and, Its Curved Surface Area $=\frac{2}{3}$ of TSA

We know that TSA of a cylinder is given by the formula $2\pi rh+2\pi r^2$, and
CSA is given by the formula $2\pi rh$, where $r$ is the radius and $h$ is the height.

According to the question, we have

$\frac{2}{3}\times$ TSA = CSA

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

$\Rightarrow \frac{2}{3}\times 2\pi r(h+r)=2\pi rh$

$\Rightarrow \frac{2}{3}\times (h+r)=h$

$\Rightarrow 2h+2r=3h$
$\therefore h=2r$

TSA $=2\pi rh+2\pi r^2=231$
$\Rightarrow 2\pi r(h+r)=231$

$\Rightarrow 2\pi r\times 3r=231$ $[\because h=2r]$

$\Rightarrow r^2=\frac{231}{6\pi }$

$\Rightarrow r^2=\frac{231}{6\times \frac{22}{7}}$

$\Rightarrow r^2=\frac{231\times 7}{6\times 22}$

$\Rightarrow r^2=\frac{21\times 7}{6\times 2}$

$\Rightarrow r^2=\frac{7\times 7}{2\times 2}$

$\Rightarrow r=\sqrt{\frac{7\times 7}{2\times 2}}$

$\Rightarrow r=\frac{7}{2}=3.5cm$

$\therefore h=2r=2\times 3.5=7cm$

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

$=\frac{22}{7}\times {3.5}^2\times 7$ [Taking $\pi =\frac{22}{7}$]

$=22\times {3.5}^2$

$=269.5c{m}^3$