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 per cent of material is wasted in this process ?

Options

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

Answer

To find the percentage of material wasted when a solid cone is melted to form a solid cylinder, we can follow these steps:

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

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

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

For the given cone:
- Radius \( r = 9 \) cm
- Height \( h = 36 \) cm

Substituting these values into the formula:

\[ V_{cone} = \frac{1}{3} \pi (9)^2 (36) \]
\[ = \frac{1}{3} \pi (81)(36) \]
\[ = \frac{1}{3} \pi (2916) \]
\[ = 972 \pi \, \text{cm}^3 \]

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

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

For the given cylinder:
- Radius \( r = 9 \) cm
- Height \( h = 9 \) cm

Substituting these values into the formula:

\[ V_{cylinder} = \pi (9)^2 (9) \]
\[ = \pi (81)(9) \]
\[ = 729 \pi \, \text{cm}^3 \]

### Step 3: Determine the volume of material wasted
To find the volume of material wasted, we subtract the volume of the cylinder from the volume of the cone:

\[ V_{wasted} = V_{cone} - V_{cylinder} \]
\[ = 972 \pi - 729 \pi \]
\[ = (972 - 729) \pi \]
\[ = 243 \pi \, \text{cm}^3 \]

### Step 4: Calculate the percentage of material wasted
To find the percentage of material wasted, we use the formula:

\[ \text{Percentage wasted} = \left( \frac{V_{wasted}}{V_{cone}} \right) \times 100 \]

Substituting the values we calculated:

\[ \text{Percentage wasted} = \left( \frac{243 \pi}{972 \pi} \right) \times 100 \]
\[ = \left( \frac{243}{972} \right) \times 100 \]
\[ = \frac{1}{4} \times 100 \]
\[ = 25\% \]

### Final Answer
Thus, the percentage of material wasted in the process is **25%**.

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