The radius of the base and the height of a cylinder are in the ratio 2 : 3. If its volume is 1617 cm3,

Suppose that the radius of the base and the height of the cylinder are 2x cm and 3x cm, respectively.

Then $1617=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times(2 \mathrm{x})^{2} \times 3 \mathrm{x}$

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

$\Rightarrow \mathrm{x}^{3}=\frac{1617 \times 7}{22 \times 12}=42.875$

$\Rightarrow \mathrm{x}=\sqrt[3]{42.875}=3.5 \mathrm{~cm}$

Hence, radius = 7 cm; height of the cylinder = 10.5 cm

$\therefore$ Total surface area of the cylinder $=2 \pi r h+2 \pi r^{2}$

$=2 \times \frac{22}{7}(7 \times 10.5+7 \times 7) \mathrm{cm}^{2}$

$=\frac{44}{7} \times(73.5+49) \mathrm{cm}^{2}=770 \mathrm{~cm}^{2}$