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

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)

$=44^{*} 17.5=770 \mathrm{~cm}^{3}$