A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter $ 42 \mathrm{~cm} $ and height $ 21 \mathrm{~cm} $ which are filled completely. Find the diameter of the cylindrical vessel.

Given:

A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter \( 42 \mathrm{~cm} \) and height \( 21 \mathrm{~cm} \) which are filled completely.

To do:

We have to find the diameter of the cylindrical vessel.

Solution:

Let $R$ be the radius of the larger cylindrical vessel.

Diameter of each small cylindrical vessel $= 42\ cm$

This implies,

Radius of each small vessel $r = \frac{42}{2} = 21\ cm$

Height of each small vessel $h = 21\ cm$

Volume of each small cylindrical vessel $= \pi r^2h$

$= \pi (21)^2\times 21$

$= 9261 \pi\ cm^3$

Volume of two small vessels $= 2 \times 9261 \pi$

$= 18522 \pi\ cm^3$

Volume of the larger cylindrical vessel $= 18522 \pi\ cm^3$

Height of the larger cylindrical vessel $H = Diameter = 2R$

Volume of the larger cylindrical vessel $=\pi R^2H$

$\Rightarrow \pi R^2 \times 2R = 18522 \pi$

$2 \pi R^3 = 18522 \pi$

$R^3 = \frac{18522}{2}$

$R^3= 9261$

$R^3 = (21)^3$

$\Rightarrow R = 21\ cm$

Diameter $= 2R$

$= 2\times21$

$=42\ cm$

The diameter of the cylindrical vessel is $42\ cm$.

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Updated on: 10-Oct-2022

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