From a wooden cubical block of side `10` cm, a sphere of radius `4.2` cm is carved out. How much wood is wasted in the process ? (Use ` pi = (22)/(7)`)

Options

A. `476.2 cm ^3`
B. `689.536 cm^3`
C. `310.464 cm^3`
D. `523.8 cm^3`

Answer

To determine how much wood is wasted when a sphere is carved out of a wooden cubical block, we need to calculate the volume of both the cube and the sphere and then find the difference between them.

### Step-by-Step Solution:

1. **Calculate the Volume of the Cube:**
The formula for the volume of a cube is given by:
\[ V_{\text{cube}} = \text{side}^3 \]
Given that the side of the cube is \(10 \, \text{cm}\):
\[ V_{\text{cube}} = 10^3 = 1000 \, \text{cm}^3 \]

2. **Calculate the Volume of the Sphere:**
The formula for the volume of a sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Given that the radius \(r\) of the sphere is \(4.2 \, \text{cm}\) and using \(\pi = \frac{22}{7}\):
\[ V_{\text{sphere}} = \frac{4}{3} \times \frac{22}{7} \times (4.2)^3 \]

3. **Calculate \( (4.2)^3 \):**
First, convert \(4.2\) to a fraction:
\[ 4.2 = \frac{42}{10} \]
Now, calculate the cube:
\[ (4.2)^3 = \left(\frac{42}{10}\right)^3 = \frac{42^3}{10^3} = \frac{74088}{1000} \]

4. **Substituting Back into the Volume of the Sphere:**
Now substitute \( (4.2)^3 \) back into the volume formula:
\[ V_{\text{sphere}} = \frac{4}{3} \times \frac{22}{7} \times \frac{74088}{1000} \]
Simplifying this:
\[ V_{\text{sphere}} = \frac{4 \times 22 \times 74088}{3 \times 7 \times 1000} \]

5. **Calculating the Numerator and Denominator:**
- Numerator: \(4 \times 22 = 88\)
- Then, \(88 \times 74088 = 6519424\)
- Denominator: \(3 \times 7 \times 1000 = 21000\)

Thus,
\[ V_{\text{sphere}} = \frac{6519424}{21000} \]

6. **Performing the Division:**
Now, divide \(6519424\) by \(21000\):
\[ V_{\text{sphere}} \approx 310.464 \, \text{cm}^3 \]

7. **Calculate the Wasted Wood:**
The wasted wood is the volume of the cube minus the volume of the sphere:
\[ \text{Wastage} = V_{\text{cube}} - V_{\text{sphere}} = 1000 - 310.464 = 689.536 \, \text{cm}^3 \]

### Final Answer:
The amount of wood wasted in the process is:
\[ \text{Wastage} = 689.536 \, \text{cm}^3 \]