The total surface area of a solid hemisphere is `108 pi cm^(2)` . The volume of the hemisphere is

Options

A. `72 pi cm^(3)`
B. `144 pi cm^(3)`
C. `108 sqrt6 cm^(3)`
D. `54 sqrt6 cm^(3)`

Answer

To solve the problem, we need to find the volume of a solid hemisphere given its total surface area. Let's go through the steps one by one.

### Step 1: Understand the formula for the total surface area of a hemisphere.
The total surface area (TSA) of a solid hemisphere is given by the formula:
\[ \text{TSA} = 3\pi r^2 \]
where \( r \) is the radius of the hemisphere.

### Step 2: Set up the equation using the given total surface area.
We are given that the total surface area of the hemisphere is \( 108\pi \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 3\pi r^2 = 108\pi \]

### Step 3: Simplify the equation.
We can divide both sides of the equation by \( \pi \):
\[ 3r^2 = 108 \]

### Step 4: Solve for \( r^2 \).
Now, divide both sides by 3:
\[ r^2 = \frac{108}{3} = 36 \]

### Step 5: Find the radius \( r \).
Taking the square root of both sides gives us:
\[ r = \sqrt{36} = 6 \, \text{cm} \]

### Step 6: Use the radius to find the volume of the hemisphere.
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3}\pi r^3 \]
Substituting \( r = 6 \, \text{cm} \) into the volume formula:
\[ V = \frac{2}{3}\pi (6)^3 \]

### Step 7: Calculate \( (6)^3 \).
Calculating \( (6)^3 \):
\[ (6)^3 = 216 \]
So, we have:
\[ V = \frac{2}{3}\pi (216) \]

### Step 8: Simplify the volume expression.
Now, simplify:
\[ V = \frac{2 \times 216}{3}\pi = \frac{432}{3}\pi = 144\pi \, \text{cm}^3 \]

### Final Answer:
The volume of the hemisphere is:
\[ \boxed{144\pi \, \text{cm}^3} \]