A cone of height 7 cm and base radius 1 cm is carved from a cuboidal block of wood `10 cm xx 5 cm xx 2 cm`. [Assuming `pi = (22)/(7)`] The percentage wood wasted in the process is :

Options

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

Answer

To find the percentage of wood wasted when carving a cone from a cuboidal block of wood, we will follow these steps:

### Step 1: Calculate the volume of the cuboidal block.
The formula for the volume of a cuboid is:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
Given dimensions are:
- Length = 10 cm
- Width = 5 cm
- Height = 2 cm

Calculating the volume:
\[ \text{Volume of cuboid} = 10 \times 5 \times 2 = 100 \text{ cm}^3 \]

### Step 2: Calculate the volume of the cone.
The formula for the volume of a cone is:
\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]
Given values are:
- Radius \( r = 1 \) cm
- Height \( h = 7 \) cm
- \( \pi = \frac{22}{7} \)

Calculating the volume of the cone:
\[ \text{Volume of cone} = \frac{1}{3} \times \frac{22}{7} \times (1^2) \times 7 \]
\[ = \frac{1}{3} \times \frac{22}{7} \times 1 \times 7 \]
\[ = \frac{1}{3} \times 22 = \frac{22}{3} \text{ cm}^3 \]

### Step 3: Calculate the volume of wood wasted.
To find the volume of wood wasted, subtract the volume of the cone from the volume of the cuboid:
\[ \text{Volume wasted} = \text{Volume of cuboid} - \text{Volume of cone} \]
\[ = 100 - \frac{22}{3} \]
To perform this subtraction, convert 100 into a fraction with a denominator of 3:
\[ 100 = \frac{300}{3} \]
Now, subtract:
\[ \text{Volume wasted} = \frac{300}{3} - \frac{22}{3} = \frac{300 - 22}{3} = \frac{278}{3} \text{ cm}^3 \]

### Step 4: Calculate the percentage of wood wasted.
The percentage of wood wasted is given by:
\[ \text{Percentage wasted} = \left( \frac{\text{Volume wasted}}{\text{Volume of cuboid}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage wasted} = \left( \frac{\frac{278}{3}}{100} \right) \times 100 \]
This simplifies to:
\[ = \frac{278}{3} \]
Calculating this gives:
\[ = 92 \frac{2}{3} \% \]

### Final Answer:
The percentage of wood wasted in the process is \( 92 \frac{2}{3} \% \).

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