Internal radius of a ball is 3 cm and external radius is 4 cm. What will be the volume of the material used.

Options

A. `(74pi)/(3)`
B. `(148pi)/(3)`
C. `(148pi)/(9)`
D. None of these

Answer

To find the volume of the material used in the spherical shell, we need to calculate the volume of the outer sphere and subtract the volume of the inner sphere.

### Step-by-Step Solution:

1. **Identify the radii**:
- Internal radius (r) = 3 cm
- External radius (R) = 4 cm

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

3. **Calculate the volume of the outer sphere**:
- Using the external radius \( R = 4 \) cm:
\[ V_{\text{outer}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

4. **Calculate the volume of the inner sphere**:
- Using the internal radius \( r = 3 \) cm:
\[ V_{\text{inner}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = \frac{108}{3} \pi \text{ cm}^3 \]

5. **Calculate the volume of the material used**:
- The volume of the material is the difference between the volume of the outer sphere and the volume of the inner sphere:
\[ V_{\text{material}} = V_{\text{outer}} - V_{\text{inner}} = \left(\frac{256}{3} \pi - \frac{108}{3} \pi\right) = \frac{256 - 108}{3} \pi = \frac{148}{3} \pi \text{ cm}^3 \]

6. **Final result**:
- The volume of the material used is:
\[ V_{\text{material}} = \frac{148}{3} \pi \text{ cm}^3 \]