A cylinder whose height is two thirds of its diameter, has the same volume as a sphere of radius $4\ cm$. Calculate the radius of the base of the cylinder.

 Given:

A cylinder whose height is two-thirds of its diameter has the same volume as a sphere of radius $4\ cm$.

To do:

We have to find the radius of the base of the cylinder.

Solution:

Radius of the sphere $(r) = 4\ cm$

This implies,

Volume of the sphere $=\frac{4}{3} \pi r^{3}$

$=\frac{4}{3} \pi \times 4 \times 4 \times 4$

$=\frac{256 \pi}{3} \mathrm{~cm}^{3}$

Therefore,

Volume of the cylinder $=\frac{256 \pi}{3} \mathrm{~cm}^{3}$

Let the diameter of the cylinder be $2R$

This implies,

Height of the cylinder $H=\frac{2}{3}(2 R)$

$=\frac{4}{3} R$

Volume $=\pi R^{2} H$

$=\pi R^{2} \times \frac{4}{3} R$

$=\frac{4}{3} \pi R^{3}$

This implies,

$\frac{4}{3} \pi R^{3}=\frac{256 \pi}{3}$

$R^{3}=\frac{256 \pi}{3} \times \frac{3}{4 \pi}$

$R^3=64$

$R^3=(4)^{3}$

$\Rightarrow R=4\ cm$

Hence, the radius of the cylinder is $4 \mathrm{~cm}$.

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Updated on: 10-Oct-2022

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