What length of a solid cylinder $ 2 \mathrm{~cm} $ in diameter must be taken to recast into a hollow cylinder of length $ 16 \mathrm{~cm} $, external diameter $ 20 \mathrm{~cm} $ and thickness $ 2.5 \mathrm{~mm} $?

Given:

Diameter of the solid cylinder $=2\ cm$

Length of the hollow cylinder $=16\ cm$

External diameter of the hollow cylinder $=20\ cm$

Thickness of the hollow cylinder $=2.5\ mm$

To do:

We have to find the length of the solid cylinder.

Solution:

Let $l$ be the length of the solid cylinder.

For hollow cylinder,

Length $= 16\ cm$

External diameter $= 20\ cm$

Thickness $= 2.5\ mm = 0.25\ cm$

External radius $R = \frac{20}{2} = 10\ cm$

Internal radius $r = 10 - 0.25 = 9.75\ cm$

Volume of the hollow cylinder $=\pi h (R^2 - r^2)$

$= \pi \times 16 (10^2 - 9.75^2)$

$= 16 \pi (100.00 - 95.0625)$

$= 16 \pi (4.9375)\ cm^3$

Volume of the solid cylinder $= 16 \pi \times 4.9375\ cm^3$

Diameter $= 2\ cm$

Radius $r_1 = \frac{2}{2} = 1\ cm$

Therefore,

$\pi r_1^2h = 16 \pi \times 4.9375$

$\pi (1)^2 h = 16 \pi  4.9375$

$h = 16 \times 4.9375$

$h= 79$

The length of the solid cylinder is $79\ cm$. 

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Updated on: 10-Oct-2022

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