# The radius of the base and the height of a solid right circular cylinder are in the ratio 2 : 3 and its volume is 1617 cm3.

Question:

The radius of the base and the height of a solid right circular cylinder are in the ratio 2 : 3 and its volume is 1617 cm3. Find the total surface area of the cylinder.

Solution:

Let the radius of the cylinder be 2x cm and its height be 3x cm.

Then, volume of the cylinder $=\pi r^{2} h$

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

Therefore,

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

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

$\Rightarrow \frac{22}{7} \times 12 x^{3}=1617$

$\Rightarrow x^{3}=\left(1617 \times \frac{7}{22} \times 12\right)$

$\Rightarrow x^{3}=\left(\frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}\right)$

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

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

Now, $r=7 \mathrm{~cm}$ and $h=\frac{21}{2} \mathrm{~cm}$

Hence, the total surface area of the cylinder:

$\left(2 \pi r h+2 \pi r^{2}\right)$

$=2 \pi r(h+r)$

$=2 \times \frac{22}{7} \times 7 \times\left(\frac{21}{2}+7\right) \mathrm{cm}^{2}$

$=\left(2 \times \frac{22}{7} \times 7 \times \frac{35}{2}\right) \mathrm{cm}^{2}$

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