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