Correct Answer: 192.5 cm ²

Solution : Given: The ratio of the radius of the base and the height of a solid right circular cylinder is 2 : 3.
Let the radius and height be  $2x$ and $3x$ respectively.
The volume is 202.125 cm ³ .
The total surface area of the cylinder = $2\pi r(r+h)$,
The volume of the cylinder = $\pi r^2h$, where $r$ and $h$ are the radius and height respectively.
⇒ $\frac{22}{7}\times (2x)^2\times 3x=202.125$
⇒ $x^3=\frac{202.125\times 7}{22\times 2\times 2\times 3}$
⇒ $x^3=\frac{0.125\times 49\times 7}{ 2\times 2\times 2}$
⇒ $x=\frac{0.5\times 7}{ 2}=1.75$
The radius of the cylinder $=2x=2\times 1.75 = 3.5$ cm
The height of the cylinder $=3x=3\times 1.75 = 5.25$ cm
The total surface area of the cylinder $=2\times\frac{22}{7}\times 3.5\times (3.5+5.25)$
$=2\times\frac{22}{7}\times 3.5 \times 8.75=192.5$ cm ²
Hence, the correct answer is 192.5 cm ² .