A solid cylinder has total surface area of $462\ cm^2$. Its curved surface area is one-third of its total surface area. Find the radius and height of the cylinder.

 Given:

A solid cylinder has total surface area of $462\ cm^2$. Its curved surface area is one-third of its total surface area. 

To do:

We have to find the radius and height of the cylinder.

Solution:

Total surface of the solid cylinder $= 462\ cm^2$

Curved surface area $=\frac{1}{3}$ of total surface area

$=\frac{1}{3} \times 462$

$=154 \mathrm{~cm}^{2}$

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

This implies,

$2 \pi r h=154$

$2 \pi r^{2}=462-154$

$=308 \mathrm{~cm}^{2}$

$\Rightarrow r^{2}=\frac{308}{2 \pi}$

$=\frac{308 \times 7}{2 \times 22}$

$=49$

$=(7)^{2}$

$\Rightarrow r=7 \mathrm{~cm}$

$2 \pi r h=154$

$r h=\frac{154}{2 \pi}$

$=\frac{154 \times 7}{2 \times 22}$

$=\frac{49}{2}$

$\Rightarrow h=\frac{49}{2 \times r}$

$=\frac{49}{2 \times 7}$

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