The total surface area of a solid hemisphere is 1039.5 `cm^2` . The volume(in `cm^3` ) of the hemisphere is: (Take `22/7`)

Options

A. 2225.5
B. 2530.6
C. 2425.5
D. 2525.6

Answer

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

### 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 TSA.
We know the total surface area is 1039.5 cm², so we can set up the equation:
\[ 3\pi r^2 = 1039.5 \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ 3 \times \frac{22}{7} r^2 = 1039.5 \]

### Step 3: Simplify the equation.
Multiply both sides by 7 to eliminate the fraction:
\[ 3 \times 22 r^2 = 1039.5 \times 7 \]
Calculating \( 1039.5 \times 7 \):
\[ 1039.5 \times 7 = 7276.5 \]
So, we have:
\[ 66 r^2 = 7276.5 \]

### Step 4: Solve for \( r^2 \).
Divide both sides by 66:
\[ r^2 = \frac{7276.5}{66} \]
Calculating \( \frac{7276.5}{66} \):
\[ r^2 = 110.25 \]

### Step 5: Find the radius \( r \).
Take the square root of both sides:
\[ r = \sqrt{110.25} = 10.5 \, \text{cm} \]

### Step 6: Calculate the volume of the hemisphere.
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
Substituting \( \pi \) with \( \frac{22}{7} \) and \( r \) with \( 10.5 \):
\[ V = \frac{2}{3} \times \frac{22}{7} \times (10.5)^3 \]

### Step 7: Calculate \( (10.5)^3 \).
Calculating \( (10.5)^3 \):
\[ (10.5)^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 \]

### Step 8: Substitute and simplify the volume formula.
Now substitute \( (10.5)^3 \) into the volume formula:
\[ V = \frac{2}{3} \times \frac{22}{7} \times 1157.625 \]

### Step 9: Calculate the volume.
First, calculate \( \frac{2}{3} \times \frac{22}{7} \):
\[ \frac{2 \times 22}{3 \times 7} = \frac{44}{21} \]
Now multiply this by \( 1157.625 \):
\[ V = \frac{44 \times 1157.625}{21} \]
Calculating \( 44 \times 1157.625 \):
\[ 44 \times 1157.625 = 50935.5 \]
Now divide by 21:
\[ V = \frac{50935.5}{21} = 2420.75 \, \text{cm}^3 \]

### Final Answer:
The volume of the hemisphere is \( 2420.75 \, \text{cm}^3 \).

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