State True or False and justify your answer
A solid ball is exactly fitted inside the cubical box of side `a`. The volume of the ball is `(4)/(3) pi a^(3)`.

Options

A. `(1)/(3) pia^(3)`
B. `(4)/(3) pia^(3)`
C. `(1)/(6) pia^(3)`
D. can not be determined

Explanation

To determine whether the statement "A solid ball is exactly fitted inside the cubical box of side `a`. The volume of the ball is `(4)/(3) pi a^(3)`" is true or false, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understand the Shapes**:
- We have a cube with side length `a`.
- Inside this cube, there is a sphere (solid ball) that is fitted exactly.

2. **Identify the Diameter and Radius of the Sphere**:
- The sphere fits perfectly inside the cube, meaning that the diameter of the sphere is equal to the side length of the cube.
- Therefore, the diameter of the sphere = `a`.
- The radius of the sphere is half of the diameter:
\[ \text{Radius} = \frac{a}{2} \]

3. **Use the Formula for Volume of a Sphere**:
- The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Substituting the radius we found:
\[ V = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 \]

4. **Calculate the Volume**:
- Calculate \( \left(\frac{a}{2}\right)^3 \):
\[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \]
- Substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi \cdot \frac{a^3}{8} \]
- Simplify:
\[ V = \frac{4 \pi a^3}{24} = \frac{\pi a^3}{6} \]

5. **Compare with the Given Statement**:
- The volume of the sphere we calculated is \( \frac{\pi a^3}{6} \).
- The statement claims the volume is \( \frac{4}{3} \pi a^3 \).
- Since \( \frac{\pi a^3}{6} \neq \frac{4}{3} \pi a^3 \), the statement is false.

### Conclusion:
The statement is **False**. The volume of the ball (sphere) that fits inside a cube of side `a` is \( \frac{\pi a^3}{6} \), not \( \frac{4}{3} \pi a^3 \).