Question

The total surface area of a solid cylinder is $462{cm}^2$. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder.

Answer 1

Let the radius of the cylinder be r and height be h.
Then Given, Total surface area $=462{cm}^2$.
$\Rightarrow 2\pi r(h+r)=464\dots$(i)
According to the question.
Curved surface area $=\frac{1}{3}($ total surface area $)$
$=\frac{1}{3}\times 462$
$=2\pi rh=154\dots \left(ii\right)$
on dividing equation (i) by equation (ii) we get.
$\begin{array}{cc}\frac{2\pi r(h+r)}{2\pi rh}& =\frac{462}{154}\\ \frac{h+r}{h}& =3\\ h+r& =3h\\ r& =3h-h\\ r& =2h\dots \left(iii\right)\end{array}$
Putting the value of $r$ from equation (iii) into equation (ii) we get,
$\begin{array}{cc}2\pi \left(2h\right)h& =154\\ 4\pi h^2& =154\\ 4\times \frac{22}{7}\times h^2& =154\\ h^2& =\frac{154\times 7}{4\times 22}=\frac{7\times 7}{4}\\ h& =\frac{7}{2}cm\\ \therefore r& =2\times \frac{7}{2}=7cm\end{array}$
$\therefore$ Volume of the cylinder $=\pi r^{2}h$
$=\frac{22}{7}\times 7\times 7\times \frac{7}{2}$
$\begin{array}{c}=\frac{22\times 7\times 7}{2}\\ =539{cm}^3\end{array}$
Hence the volume of cylinder $=539{cm}^3$