The volume of a solid hemisphere is `19404 cm^(3)`. Its total surface area is

Options

A. `4158 cm^(2)`
B. `2858 cm^(2)`
C. `1738 cm^(2)`
D. `2038 cm^(2)`

Answer

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

### Step 1: Use the formula for the volume of a hemisphere
The volume \( V \) of a solid hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi R^3 \]
where \( R \) is the radius of the hemisphere.

### Step 2: Set the volume equal to the given volume
We know the volume of the hemisphere is \( 19404 \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ \frac{2}{3} \pi R^3 = 19404 \]

### Step 3: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we can substitute this value into the equation:
\[ \frac{2}{3} \times \frac{22}{7} R^3 = 19404 \]

### Step 4: Simplify the equation
To eliminate the fractions, we can multiply both sides by \( 3 \times 7 \):
\[ 2 \times 22 R^3 = 19404 \times 3 \times 7 \]
Calculating the right side:
\[ 19404 \times 3 = 58212 \]
\[ 58212 \times 7 = 407484 \]
Thus, we have:
\[ 44 R^3 = 407484 \]

### Step 5: Solve for \( R^3 \)
Now, divide both sides by 44:
\[ R^3 = \frac{407484}{44} \]
Calculating this gives:
\[ R^3 = 9261 \]

### Step 6: Find \( R \)
Now, take the cube root of both sides to find \( R \):
\[ R = \sqrt[3]{9261} = 21 \, \text{cm} \]

### Step 7: Calculate the total surface area
The total surface area \( A \) of a hemisphere is given by the formula:
\[ A = 3 \pi R^2 \]
Substituting \( R = 21 \, \text{cm} \):
\[ A = 3 \times \frac{22}{7} \times (21)^2 \]

### Step 8: Calculate \( R^2 \)
Calculating \( R^2 \):
\[ R^2 = 21^2 = 441 \]

### Step 9: Substitute \( R^2 \) back into the surface area formula
Now substituting \( R^2 \):
\[ A = 3 \times \frac{22}{7} \times 441 \]

### Step 10: Simplify the calculation
Calculating \( 3 \times 441 = 1323 \):
\[ A = \frac{22 \times 1323}{7} \]
Calculating \( 22 \times 1323 = 29106 \):
\[ A = \frac{29106}{7} = 4158 \]

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

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