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.
Question
The radius of the base and the height of a cylinder are in the ratio 2:3. If its volume is 1617 cm³, find the total surface area of the cylinder.
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Answer
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.

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

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, $$ \frac{22}{7} \times 4x^2 \times 3x = 1617 $$ $$ \frac{22}{7} \times 12x^3 = 1617 $$ $$ \frac{264x^3}{7} = 1617 $$ $$ 264x^3 = 1617 \times 7 $$ $$ 264x^3 = 11319 $$ $$ x^3 = \frac{11319}{264} $$ $$ x^3 = 42.8864 $$

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

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

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.
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