If the surface area of a sphere is 616 `cm^(2),` find its volume.

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Value**:
- The surface area of the sphere is given as \( 616 \, \text{cm}^2 \).

2. **Use the Formula for Surface Area of a Sphere**:
- The formula for the surface area \( A \) of a sphere is:
\[ A = 4\pi r^2 \]
- Substituting the given surface area into the formula:
\[ 4\pi r^2 = 616 \]

3. **Substitute the Value of \(\pi\)**:
- We can use \( \pi \approx \frac{22}{7} \):
\[ 4 \times \frac{22}{7} \times r^2 = 616 \]

4. **Simplify the Equation**:
- Multiply \( 4 \times \frac{22}{7} \):
\[ \frac{88}{7} r^2 = 616 \]

5. **Cross Multiply to Solve for \( r^2 \)**:
- Rearranging gives:
\[ r^2 = 616 \times \frac{7}{88} \]

6. **Calculate \( 616 \div 88 \)**:
- This simplifies to:
\[ 616 \div 88 = 7 \]
- Therefore:
\[ r^2 = 7 \times 7 = 49 \]

7. **Find the Radius \( r \)**:
- Taking the square root gives:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

8. **Use the Formula for Volume of a Sphere**:
- The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
- Substituting \( r = 7 \, \text{cm} \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times 7^3 \]

9. **Calculate \( 7^3 \)**:
- \( 7^3 = 343 \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times 343 \]

10. **Simplify the Volume Calculation**:
- The \( 7 \) in the denominator cancels with one \( 7 \) in the numerator:
\[ V = \frac{4 \times 22 \times 49}{3} \]
- Calculate \( 4 \times 22 = 88 \):
\[ V = \frac{88 \times 49}{3} \]

11. **Final Calculation**:
- \( 88 \times 49 = 4312 \):
\[ V = \frac{4312}{3} \approx 1437.33 \, \text{cm}^3 \]

12. **Conclusion**:
- The volume of the sphere is:
\[ \text{Volume} = 1437.33 \, \text{cm}^3 \]