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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 $4 \mathrm{~cm}$.