Question:
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 $1617 \mathrm{~cm}^{2}$.
Solution:
Let, r be the radius of the cylinder
h be the height of the cylinder
r/h = 2/3
r = 2/3 * h ....1
Volume of cylinder $=\pi r^{2}{ }^{*} \mathrm{~h}$
$1617=22 / 7 *(2 / 3 * h)^{2} * h$
$1617=22 / 7 *(2 / 3 * h)^{3}$
$\mathrm{h}^{3}=\frac{1617 * 7 * 3}{22 * 4}$
$h=\frac{3 * 7}{2}$
h = 10.5 cm
from, eq 1
r = 2/3 * 10.5
= 7 cm
Total surface area of cylinder = 2πr (h + r)
= 2 * 22/7 * 7(10.5 + 7)
$=44^{*} 17.5=770 \mathrm{~cm}^{3}$