The radius of the base and the height of a cylinder are in the ratio 2:3. If its volume is 1617 cm^3, find the total surface area of the cylinder.

Given: The radius of the base and the height of a cylinder are in the ratio 2:3. The volume of the cylinder is 1617 cm³. We need to find the total surface area of the cylinder.

1. Let the radius $r$ and the height $h$ of the cylinder be $2x$ cm and $3x$ cm respectively.

2. Volume of the cylinder is given by: $$V = \pi r^2 h = 1617 , \text{cm}^3$$ Substituting the values, $$\pi (2x)^2 (3x) = 1617 , \text{cm}^3$$ Simplifying further, $$\frac{22}{7} \times 4x^2 \times 3x = 1617$$ $$\frac{22}{7} \times 12x^3 = 1617$$ $$\frac{264x^3}{7} = 1617$$ $$264x^3 = 1617 \times 7$$ $$264x^3 = 11319$$ $$x^3 = \frac{11319}{264}$$ $$x^3 = 42.8864$$

3. Taking the cube root of both sides: $$x \approx 3.5$$

4. Therefore:

   • The radius $r$ is: $$2x = 2 \times 3.5 = 7 , \text{cm}$$
   • The height $h$ is: $$3x = 3 \times 3.5 = 10.5 , \text{cm}$$

5. The total surface area of the cylinder is given by: $$\text{TSA} = 2\pi r (r + h)$$ Substituting the values, $$\text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 10.5)$$ $$\text{TSA} = 2 \times 22 \times 17.5$$ $$\text{TSA} = 770 , \text{cm}^2$$

Therefore, the total surface area of the cylinder is 770 cm².