The difference between inside and outside surface of a cylindrical tube $14 c m$ long is $88 s q . c m$ . If the volume of the tube is $176$ cubic cm, find the inner diameter of the tube.(in cm)

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Solution

Given: Let $r_{1}$ be the radius of inner tube, $r_{2}$ be the radius of outer tube
$\Rightarrow 2 π r_{2} h - 2 π r_{1} h = 88$ sq. cm
$\Rightarrow 2 π (r_{2} - r_{1}) h = 88$
$\Rightarrow 2 \times \frac{22}{7} (r_{2} - r_{1}) \times 14 = 88$
$\Rightarrow r_{2} - r_{1} = 1$ …………. $(1)$
Volume of the tube $=$ volume of outer tube $-$ volume of inner tube
$= π r_{2}^{2} h - π r_{1}^{2} h$
$= π (r_{2}^{2} - r_{1}^{2}) h = π (r_{2} - r_{1}) (r_{2} + r_{1}) h$ from $(1)$
$4 = π (r_{1} + r_{2}) \times 14$
$\Rightarrow 176 = \frac{22}{7} (r_{1} + r_{2}) \times 14$
$\Rightarrow r_{1} + r_{2} = 4$ …….. $(2)$
From $(1)$ & $(2)$
$r_{1} + r_{2} + r_{2} - r_{1} = 4 + 1$
$\Rightarrow 2 r_{2} = 5$
$\Rightarrow r_{2} = \frac{5}{2}$
$= 2.5$ cm
From $(2)$
$r_{1} = 4 - r_{2} = (4 - 2.5) c m$
$= 1.5 c m$
$∴$ Inner radius $r_{1} = 1.5$ cm
Inner diameter $= 2 r_{1} = 2 \times 1.5$
$= 3$ cm.

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The difference between inside and outside surface of a cylindrical tube 14cm long is 88sq.cm. If the volume of the tube is 176 cubic cm, find the inner diameter of the tube.(in cm)

The difference between inside and outside surface of a cylindrical tube $14 c m$ long is $88 s q . c m$ . If the volume of the tube is $176$ cubic cm, find the inner diameter of the tube.(in cm)