A hollow spherical metallic bal has an external diameter 6 cm and is `(1)/(2) cm` thick. The volume of the ball (in `cm^(3)`) is (Take `pi = (22)/(7)`)

Options

A. `41(2)/(3)`
B. 207.868
C. `47(2)/(3)`
D. `40(2)/(3)`

Answer

To find the volume of a hollow spherical metallic ball, we need to follow these steps:

### Step 1: Determine the External and Internal Radius
The external diameter of the ball is given as 6 cm. Therefore, the external radius (R1) can be calculated as:
\[ R1 = \frac{\text{External Diameter}}{2} = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm} \]

The thickness of the ball is given as \( \frac{1}{2} \, \text{cm} \). Thus, the internal radius (R2) can be calculated by subtracting the thickness from the external radius:
\[ R2 = R1 - \text{Thickness} = 3 \, \text{cm} - \frac{1}{2} \, \text{cm} = 3 \, \text{cm} - 0.5 \, \text{cm} = 2.5 \, \text{cm} \]

### Step 2: Calculate the Volume of the Hollow Sphere
The volume of a hollow sphere is given by the formula:
\[ V = \frac{4}{3} \pi (R1^3 - R2^3) \]

Substituting the values of R1 and R2 into the formula:
\[ V = \frac{4}{3} \times \frac{22}{7} \left(3^3 - (2.5)^3\right) \]

Calculating \( R1^3 \) and \( R2^3 \):
- \( R1^3 = 3^3 = 27 \)
- \( R2^3 = (2.5)^3 = 15.625 \)

Now, substituting these values:
\[ V = \frac{4}{3} \times \frac{22}{7} \left(27 - 15.625\right) \]
\[ V = \frac{4}{3} \times \frac{22}{7} \times 11.375 \]

### Step 3: Simplify the Volume Calculation
Now we calculate \( 27 - 15.625 \):
\[ 27 - 15.625 = 11.375 \]

Next, we calculate:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 11.375 \]

Calculating \( \frac{4 \times 22 \times 11.375}{3 \times 7} \):
1. Calculate \( 4 \times 22 = 88 \)
2. Calculate \( 88 \times 11.375 = 1000.5 \)
3. Calculate \( 3 \times 7 = 21 \)

Now, we have:
\[ V = \frac{1000.5}{21} \]

### Step 4: Final Calculation
Calculating \( \frac{1000.5}{21} \):
\[ V \approx 47.619 \, \text{cm}^3 \]

### Final Answer
Thus, the volume of the hollow spherical metallic ball is approximately:
\[ V \approx 47.62 \, \text{cm}^3 \]