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The difference between the outer and inner curved surface areas of a hollow right circular cylinder $ 14 \mathrm{~cm} $ long is $ 88 \mathrm{~cm}^{2} $. If the volume of metal used in making the cylinder is $ 176 \mathrm{~cm}^{3} $, find the outer and inner diameters of the cylinder. (Use $ \pi=22 / 7 $ )
Given:
The difference between the outer and inner curved surface areas of a hollow right circular cylinder \( 14 \mathrm{~cm} \) long is \( 88 \mathrm{~cm}^{2} \).
The volume of metal used in making the cylinder is \( 176 \mathrm{~cm}^{3} \).
To do:
We have to find the outer and inner diameters of the cylinder.
Solution:
Height of the hollow right circular cylinder $= 14\ cm$
Difference between the outer and inner curved surface areas $= 88\ cm^2$
Volume of the metal used in making the cylinder $=176\ cm^3$
Let $\mathrm{R}$ and $r$ be the outer and inner radii of the cylinder.
$\Rightarrow \pi R^{2} h-\pi r^{2} h=176$
$\Rightarrow \pi h(R^{2}-r^{2})=176$
$\Rightarrow \frac{22}{7} \times 14(\mathrm{R}^{2}-r^{2})=176$
$\Rightarrow \mathrm{R}^{2}-r^{2}=\frac{176 \times 7}{22 \times 14}$
$\Rightarrow \mathrm{R}^{2}-r^{2}=4 \).........(i)
$2 \pi \mathrm{R} h-2 \pi r h=88$
$\Rightarrow 2 \pi h(\mathrm{R}-r)=88$
$\Rightarrow 2 \times \frac{22}{7} \times 14(\mathrm{R}-r)=88$
$\Rightarrow \mathrm{R}-r=\frac{88 \times 7}{2 \times 22 \times 14}$
$\Rightarrow \mathrm{R}-r=1$.........(ii)
Therefore,
$\mathrm{R}^{2}-r^{2}=4$
$\Rightarrow (\mathrm{R}+r)(\mathrm{R}-r)=4$
$\Rightarrow (\mathrm{R}+r)(1)=4$ [From (ii)]
$\Rightarrow \mathrm{R}+r=4$............(iii)
Adding equations (ii) and (iii), we get,
$2 \mathrm{R}=5$
$\mathrm{R}=\frac{5}{2} \mathrm{~cm}$
Substituting $\mathrm{R}=\frac{5}{2} \mathrm{~cm}$ in (ii), we get,
$\frac{5}{2}-r=1$
$\Rightarrow r=\frac{5}{2}-1$
$\Rightarrow r=\frac{3}{2} \mathrm{~cm}$
Therefore,
Outer diameter of the cylinder $=\frac{5}{2} \times 2=5 \mathrm{~cm}$
Inner diameter of the cylinder $=\frac{3}{2} \times 2=3 \mathrm{~cm}$
The outer and inner diameters of the cylinder are $5\ cm$ and $3\ cm$ respectively.
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