If the curved surface area of a hemisphere is $198\pi$ cm², calculate its volume in cm³.

The correct answer is $\frac{198\sqrt{99}}{3}\pi$.

Given:
Curved surface area (CSA) of the hemisphere = $198\pi$ cm²

Formulae:
1. Curved Surface Area of a hemisphere = $2\pi r^2$
2. Volume of a hemisphere = $\frac{2}{3}\pi r^3$
where $r$ represents the radius of the hemisphere.

Step 1: Find the radius ($r$) of the hemisphere.

Equating the given curved surface area to its formula:

$2\pi r^2=198\pi$

Divide both sides of the equation by $\pi$:

$2r^2=198$

Divide both sides by 2 to solve for $r^2$:

$r^2=99$

Taking the square root on both sides:

$r=\sqrt{99}$ cm

Step 2: Calculate the volume ($V$) of the hemisphere.

Substitute the expressions for $r^2$ and $r$ into the volume formula:

$V=\frac{2}{3}\pi r^3$

We can rewrite $r^3$ as $r^2\times r$:

$V=\frac{2}{3}\pi \times 99\times \sqrt{99}$

Multiplying the values in the numerator:

$2\times 99=198$

Placing this back into the formula gives:

$V=\frac{198\sqrt{99}}{3}\pi$ cm³

Therefore, the volume of the hemisphere is $\frac{198\sqrt{99}}{3}\pi$ cm³.