The ratio between the curved surface area and the total surface area of a right circular cylinder is $1 : 2$. Find the volume of the cylinder, if its total surface area is $616\ cm^2$.

 Given:

The ratio between the curved surface area and the total surface area of a right circular cylinder is $1 : 2$.

Total surface area is $616\ cm^2$.

To do:

We have to find the volume of the cylinder.

Solution:

Ratio in the curved surface area and total surface area of the cylinder $=1:2$

Total surface area $= 616\ cm^2$

Therefore,

Curved surface area $=\frac{616 \times 1}{2}$

$=308 \times 1$

$=308 \mathrm{~cm}^{2}$

This implies,

$2 \pi r h=308$

$\frac{2 \times 22}{7} r h=308$

$r h=\frac{308 \times 7}{2 \times 22}$

$rh=49$

$2 \pi r^{2}=616-308$

$=308 \mathrm{~cm}^{2}$

$2 \times \frac{22}{7} r^{2}=308$

$r^{2}=\frac{308 \times 7}{2 \times 22}$

$=49$

$=(7)^{2}$

$\Rightarrow r=7 \mathrm{~cm}$

$h=\frac{49}{r}$

$=\frac{49}{7}$

$=7 \mathrm{~cm}$

Therefore,

Volume $=\pi r^{2} h$

$=\frac{22}{7} \times 7 \times 7 \times 7$

$=1078 \mathrm{~cm}^{3}$