# The ratio between the radius of the base and the height of a cylinder is $2 : 3$. Find the total surface area of the cylinder, if its volume is $1617\ cm^3$.

Given:

The ratio between the radius of the base and the height of a cylinder is $2 : 3$.

The volume is $1617\ cm^3$.

To do:

We have to find the total surface area of the cylinder.

Solution:

Ratio between radius and height of the cylinder $= 2:3$

Volume $=1617\ cm^3$

Let the radius of the cylinder be $(r) = 2x$

This implies,

Height of the cylinder $(h) = 3x$

Volume of the cylinder $= \pi r^2h$

$\frac{22}{7} \times(2 x)^{2} \times 3 x=1617$

$\frac{22}{7} \times 4 x^{2} \times 3 x=1617$

$x^{3}=\frac{1617 \times 7}{22 \times 4 \times 3}$

$x^3=\frac{343}{8}$

$x^{3}=(\frac{7}{2})^{3}$

$\Rightarrow x=\frac{7}{2}$

Therefore,

Radius $=2 x$

$=2 \times \frac{7}{2}$

$=7 \mathrm{~cm}$

Height $=3 x$

$=3 \times \frac{7}{2}$

$=\frac{21}{2} \mathrm{~cm}$

Total surface area $=2 \pi r(h+r)$

$=2 \times \frac{22}{7} \times 7(\frac{21}{2}+7)$

$=44 \times(\frac{21+14}{2})$

$=44 \times \frac{35}{2} \mathrm{~cm}^{2}$

$=770 \mathrm{~cm}^{2}$

The total surface area of the cylinder is $770 \mathrm{~cm}^{2}$.

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Updated on: 10-Oct-2022

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