The difference between inside and outside surfaces of a cylindrical tube $14\ cm$ long is $88\ sq.\ cm$. If the volume of the tube is $176$ cubic cm, find the inner and outer radii of the tube.

 Given:

The difference between inside and outside surfaces of a cylindrical tube $14\ cm$ long is $88\ sq.\ cm$.

The volume of the tube is $176$ cubic cm.

To do:

We have to find the inner and outer radii of the tube.

Solution:

Length of the cylindrical tube $= 14\ cm$

Difference between the outer surface and inner surface $= 88\ cm^2$

Volume of the tube $= 176\ cm^3$

Let $R$ and $r$ be the outer and inner radii of the tube.

Therefore,

$2 \pi \mathrm{R} h-2 \pi r h=88$

$2 \pi h(\mathrm{R}-r)=88$

$2 \times \frac{22}{7} \times 14(\mathrm{R}-r)=88$

$\mathrm{R}-r=\frac{88 \times 7}{2 \times 22 \times 14}$

$\mathrm{R}-r=1 \mathrm{~cm}$..........(i)

Volume of the tube $=176$

$\pi \mathrm{R}^{2} h-\pi r^{2} h=176$

$\pi h(\mathrm{R}^{2}-r^{2})=176$

$\frac{22}{7} \times 14(\mathrm{R}^{2}-r^{2})=176$

$\mathrm{R}^{2}-r^{2}=\frac{176 \times 7}{22 \times 14}$

$\Rightarrow (\mathrm{R}+r)(\mathrm{R}-r)=4$.............(ii)

Dividing (ii) by (i), we get,

$\mathrm{R}+r=4$.........(iii)

$\mathrm{R}-r=1$..........(iv)

Adding (iii) and (iv), we get,

$2 R=5$

$R=\frac{5}{2}$

$R=2.5 \mathrm{~cm}$

This implies,

$2 r=3$

$r=\frac{3}{2}$

$r=1.5 \mathrm{~cm}$

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Updated on: 10-Oct-2022

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