The surface area of a sphere is `2464 cm^(2),` find its volume.

Answer

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

### Step 1: Write the formula for the surface area of a sphere.
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set up the equation with the given surface area.
We know the surface area is \( 2464 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 4 \pi r^2 = 2464 \]

### Step 3: Substitute the value of \( \pi \).
Using \( \pi \approx \frac{22}{7} \), we can rewrite the equation:
\[ 4 \times \frac{22}{7} \times r^2 = 2464 \]

### Step 4: Simplify the equation.
To eliminate the fraction, multiply both sides by \( 7 \):
\[ 4 \times 22 \times r^2 = 2464 \times 7 \]
Calculating \( 2464 \times 7 \):
\[ 2464 \times 7 = 17248 \]
So we have:
\[ 88 r^2 = 17248 \]

### Step 5: Solve for \( r^2 \).
Now, divide both sides by \( 88 \):
\[ r^2 = \frac{17248}{88} \]
Calculating \( \frac{17248}{88} \):
\[ r^2 = 196 \]

### Step 6: Find the radius \( r \).
Taking the square root of both sides:
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Step 7: Write the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 8: Substitute the value of \( r \).
Now substitute \( r = 14 \, \text{cm} \) into the volume formula:
\[ V = \frac{4}{3} \times \frac{22}{7} \times (14)^3 \]

### Step 9: Calculate \( (14)^3 \).
Calculating \( (14)^3 \):
\[ 14^3 = 2744 \]
So we have:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 2744 \]

### Step 10: Simplify the expression.
First, simplify \( \frac{22}{7} \times 2744 \):
\[ \frac{22 \times 2744}{7} = 22 \times 392 = 8644 \]
Now substitute back into the volume formula:
\[ V = \frac{4}{3} \times 8644 \]

### Step 11: Calculate the final volume.
Calculating \( \frac{4 \times 8644}{3} \):
\[ V = \frac{34576}{3} \approx 11525.33 \, \text{cm}^3 \]

### Final Answer:
The volume of the sphere is approximately \( 11525.33 \, \text{cm}^3 \).

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