The volume of a sphere is 4851 `cm^(3)`. Find its diameter.

Answer

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

### Step-by-Step Solution

1. **Identify the given volume of the sphere.**
- The volume \( V \) of the sphere is given as \( 4851 \, \text{cm}^3 \).

2. **Write down the formula for the volume of a sphere.**
- The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
- Here, \( r \) is the radius of the sphere.

3. **Substitute the known volume into the formula.**
- We can set up the equation:
\[ \frac{4}{3} \pi r^3 = 4851 \]

4. **Substitute the value of \( \pi \).**
- Using \( \pi \approx \frac{22}{7} \), we substitute this into the equation:
\[ \frac{4}{3} \times \frac{22}{7} \times r^3 = 4851 \]

5. **Clear the fraction by multiplying both sides by \( \frac{3 \times 7}{4 \times 22} \).**
- Rearranging gives:
\[ r^3 = 4851 \times \frac{3 \times 7}{4 \times 22} \]

6. **Calculate the right-hand side.**
- First, calculate \( 4 \times 22 = 88 \).
- Then, calculate \( 3 \times 7 = 21 \).
- Now, we have:
\[ r^3 = 4851 \times \frac{21}{88} \]

7. **Perform the multiplication and division.**
- Calculate \( 4851 \div 11 = 441 \) (since both are multiples of 11).
- Now, we can rewrite:
\[ r^3 = 441 \times \frac{21}{8} \]

8. **Find the cube root of \( r^3 \).**
- Since \( 441 = 21 \times 21 \) and \( 21 = 3 \times 7 \), we can express \( r^3 \) as:
\[ r^3 = \frac{21 \times 21 \times 21}{2 \times 2 \times 2} \]
- Thus, \( r = \frac{21}{2} = 10.5 \, \text{cm} \).

9. **Calculate the diameter.**
- The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \times 10.5 = 21 \, \text{cm} \]

10. **State the final answer.**
- The diameter of the sphere is \( 21 \, \text{cm} \).