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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,
$2 \pi r h=154$
$2 \pi r^{2}=462-154$
$=308 \mathrm{~cm}^{2}$
$\Rightarrow r^{2}=\frac{308}{2 \pi}$
$=\frac{308 \times 7}{2 \times 22}$
$=49$
$=(7)^{2}$
$\Rightarrow r=7 \mathrm{~cm}$
$2 \pi r h=154$
$r h=\frac{154}{2 \pi}$
$=\frac{154 \times 7}{2 \times 22}$
$=\frac{49}{2}$
$\Rightarrow h=\frac{49}{2 \times r}$
$=\frac{49}{2 \times 7}$
$=\frac{7}{2}\ cm$