The volume of a hemisphere is 19404 `cm^3`. The total surface area of the hemisphere is

Options

A. `4158 cm^(2)`
B. `16632 cm^(2)`
C. `8316 cm^(2)`
D. `3696 cm^(2)`

Explanation

To find the total surface area of a hemisphere given its volume, we can follow these steps:

### Step 1: Write down the formula for the volume of a hemisphere.
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]

### Step 2: Substitute the given volume into the formula.
We know the volume \( V \) is 19404 cm³. Therefore, we can set up the equation:
\[ 19404 = \frac{2}{3} \pi r^3 \]

### Step 3: Substitute the value of \( \pi \).
For simplicity, we can use \( \pi \approx \frac{22}{7} \). Substituting this into the equation gives:
\[ 19404 = \frac{2}{3} \times \frac{22}{7} \times r^3 \]

### Step 4: Solve for \( r^3 \).
To isolate \( r^3 \), we first multiply both sides by \( \frac{3 \times 7}{2 \times 22} \):
\[ r^3 = 19404 \times \frac{3 \times 7}{2 \times 22} \]

Calculating the right side:
\[ r^3 = 19404 \times \frac{21}{44} \]

### Step 5: Simplify the calculation.
Calculating \( 19404 \times 21 \):
\[ 19404 \times 21 = 407484 \]
Now divide by 44:
\[ r^3 = \frac{407484}{44} = 9249 \]

### Step 6: Find the cube root to get \( r \).
Now, we need to find \( r \):
\[ r = \sqrt[3]{9249} \]
Calculating the cube root gives:
\[ r = 21 \text{ cm} \]

### Step 7: Calculate the total surface area of the hemisphere.
The total surface area \( A \) of a hemisphere is given by:
\[ A = 3 \pi r^2 \]
Substituting \( r = 21 \) cm and \( \pi \approx \frac{22}{7} \):
\[ A = 3 \times \frac{22}{7} \times (21)^2 \]
Calculating \( (21)^2 = 441 \):
\[ A = 3 \times \frac{22}{7} \times 441 \]

### Step 8: Simplify the total surface area calculation.
Calculating \( 3 \times 22 = 66 \):
\[ A = \frac{66 \times 441}{7} \]
Calculating \( 66 \times 441 = 29046 \):
\[ A = \frac{29046}{7} = 4158 \text{ cm}^2 \]

### Final Answer:
The total surface area of the hemisphere is \( 4158 \text{ cm}^2 \).

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