A solid cone of height 36 cm and radius of base 9 cm is melted to form a solid cylinder of radius 9 cm and height 9 cm. What percent of material is wasted in this process?

Options

A. 25
B. 0
C. 5
D. 10

Answer

To solve the problem step by step, we need to calculate the volumes of the solid cone and the solid cylinder, then determine the amount of material wasted during the melting process, and finally calculate the percentage of the material that was wasted.

### 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
- \( h \) is the height

Given:
- Height \( h = 36 \) cm
- Radius \( r = 9 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (9)^2 (36) \]
Calculating \( (9)^2 = 81 \):
\[ V = \frac{1}{3} \pi (81)(36) \]
Calculating \( 81 \times 36 = 2916 \):
\[ V = \frac{1}{3} \pi (2916) \]
Calculating \( \frac{2916}{3} = 972 \):
\[ V = 972 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the base
- \( h \) is the height

Given:
- Height \( h = 9 \) cm
- Radius \( r = 9 \) cm

Substituting the values into the formula:
\[ V = \pi (9)^2 (9) \]
Calculating \( (9)^2 = 81 \):
\[ V = \pi (81)(9) \]
Calculating \( 81 \times 9 = 729 \):
\[ V = 729 \pi \text{ cm}^3 \]

### Step 3: Calculate the amount of material wasted
The amount of material wasted is the difference between the volume of the cone and the volume of the cylinder:
\[ \text{Wastage} = \text{Volume of Cone} - \text{Volume of Cylinder} \]
Substituting the volumes:
\[ \text{Wastage} = 972 \pi - 729 \pi = (972 - 729) \pi = 243 \pi \text{ cm}^3 \]

### Step 4: Calculate the percentage of material wasted
The percentage of material wasted is calculated using the formula:
\[ \text{Percentage Wasted} = \left( \frac{\text{Wastage}}{\text{Volume of Cone}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Wasted} = \left( \frac{243 \pi}{972 \pi} \right) \times 100 \]
Cancelling \( \pi \):
\[ \text{Percentage Wasted} = \left( \frac{243}{972} \right) \times 100 \]
Calculating \( \frac{243}{972} = \frac{1}{4} \):
\[ \text{Percentage Wasted} = \frac{1}{4} \times 100 = 25\% \]

### Final Answer
The percentage of material wasted in this process is **25%**.