A cone of height 7 cm and base radius 1 cm is curved froma conical block of wood `10 m xx 5 xx 2 ` cm . The percentage wood wastage in the process is `(pi = (22)/(7))`

Options

A. `92 (2)/(3)%`
B. `46 (1)/(3)%`
C. `53 (2)/(3)%`
D. `7(1)/(3)%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Volume of the Cone
The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- Radius \( r = 1 \) cm,
- Height \( h = 7 \) cm,
- \( \pi = \frac{22}{7} \).

Substituting the values into the formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (1)^2 \times 7 \]

Calculating this:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 1 \times 7 = \frac{22}{3} \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Cuboidal Block
The volume \( V \) of a cuboid is given by:

\[ V = \text{length} \times \text{width} \times \text{height} \]

Given the dimensions of the cuboidal block are \( 10 \text{ m} \times 5 \text{ cm} \times 2 \text{ cm} \). First, we need to convert the length from meters to centimeters:

\[ 10 \text{ m} = 1000 \text{ cm} \]

Now substituting the values:

\[ V = 1000 \text{ cm} \times 5 \text{ cm} \times 2 \text{ cm} = 10000 \text{ cm}^3 \]

### Step 3: Calculate the Wood Wastage
The wastage of wood is calculated by subtracting the volume of the cone from the volume of the cuboidal block:

\[ \text{Wastage} = \text{Volume of Cuboidal Block} - \text{Volume of Cone} \]

Substituting the values we calculated:

\[ \text{Wastage} = 10000 \text{ cm}^3 - \frac{22}{3} \text{ cm}^3 \]

To perform this subtraction, we convert \( 10000 \) into a fraction with a denominator of \( 3 \):

\[ 10000 = \frac{30000}{3} \]

Now, substituting this into the wastage formula:

\[ \text{Wastage} = \frac{30000}{3} - \frac{22}{3} = \frac{30000 - 22}{3} = \frac{29978}{3} \text{ cm}^3 \]

### Step 4: Calculate the Percentage of Wood Wastage
The percentage of wood wastage is given by:

\[ \text{Percentage Wastage} = \left( \frac{\text{Wastage}}{\text{Volume of Cuboidal Block}} \right) \times 100\% \]

Substituting the values:

\[ \text{Percentage Wastage} = \left( \frac{\frac{29978}{3}}{10000} \right) \times 100\% \]

Calculating this:

\[ = \left( \frac{29978}{30000} \right) \times 100\% \approx 99.926\% \]

### Final Answer
The percentage of wood wastage in the process is approximately \( 99.93\% \).