A solid cone of height 24 cm and having radius of base 8 cm is melted to form a solid cylinder of radius 6 cm and height 6 cm. In the whole process what percent of material is wasted?

Options

A. 48.5
B. 37.5
C. 57.8
D. 64

Answer

To solve the problem step by step, we will calculate the volume of the cone, the volume of the cylinder, and then determine the percentage of material wasted during the process.

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

\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

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

Given:
- Radius \( r = 8 \) cm
- Height \( h = 24 \) cm

Substituting the values:

\[ V_{\text{cone}} = \frac{1}{3} \pi (8^2) (24) = \frac{1}{3} \pi (64) (24) \]

Calculating \( 64 \times 24 \):

\[ 64 \times 24 = 1536 \]

Now substituting back:

\[ V_{\text{cone}} = \frac{1}{3} \pi (1536) = 512 \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Cylinder
The formula for the volume of a cylinder is given by:

\[ V_{\text{cylinder}} = \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cylinder
- \( h \) is the height of the cylinder

Given:
- Radius \( r = 6 \) cm
- Height \( h = 6 \) cm

Substituting the values:

\[ V_{\text{cylinder}} = \pi (6^2) (6) = \pi (36) (6) = 216 \pi \text{ cm}^3 \]

### Step 3: Calculate the Wastage
To find the wastage, we subtract the volume of the cylinder from the volume of the cone:

\[ \text{Wastage} = V_{\text{cone}} - V_{\text{cylinder}} = 512 \pi - 216 \pi = 296 \pi \text{ cm}^3 \]

### Step 4: Calculate the Percentage of Material Wasted
The percentage of material wasted can be calculated using the formula:

\[ \text{Percentage of wastage} = \left( \frac{\text{Wastage}}{\text{Volume of the cone}} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage of wastage} = \left( \frac{296 \pi}{512 \pi} \right) \times 100 \]

The \( \pi \) cancels out:

\[ \text{Percentage of wastage} = \left( \frac{296}{512} \right) \times 100 \]

Calculating \( \frac{296}{512} \):

\[ \frac{296}{512} \approx 0.578125 \]

Now multiplying by 100:

\[ \text{Percentage of wastage} \approx 57.8125\% \]

Thus, the percentage of material wasted is approximately **57.8%**.

### Final Answer
The percentage of material wasted is **57.8%**.