A solid cone of height 24 cm and havig 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 per cent of material is wasted ?

Options

A. a) `48.5`
B. b) `37.5`
C. c) `57.8`
D. d) 64

Answer

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

### 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:
- \( r \) = radius of the base of the cone = 8 cm
- \( h \) = height of the cone = 24 cm
- \( \pi \approx \frac{22}{7} \)

Substituting the values into the formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (8)^2 \times 24 \]

Calculating \( (8)^2 = 64 \):

\[ V = \frac{1}{3} \times \frac{22}{7} \times 64 \times 24 \]

Calculating \( 64 \times 24 = 1536 \):

\[ V = \frac{1}{3} \times \frac{22}{7} \times 1536 \]

Calculating \( \frac{22 \times 1536}{3 \times 7} \):

\[ V = \frac{33792}{21} \approx 1609.14 \, \text{cm}^3 \]

### 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:
- \( r \) = radius of the base of the cylinder = 6 cm
- \( h \) = height of the cylinder = 6 cm

Substituting the values into the formula:

\[ V = \frac{22}{7} \times (6)^2 \times 6 \]

Calculating \( (6)^2 = 36 \):

\[ V = \frac{22}{7} \times 36 \times 6 \]

Calculating \( 36 \times 6 = 216 \):

\[ V = \frac{22 \times 216}{7} \]

Calculating \( \frac{4752}{7} \approx 678.86 \, \text{cm}^3
\]

### Step 3: Calculate the Volume of Material Wasted
Now, we will find the volume of material wasted by subtracting the volume of the cylinder from the volume of the cone:

\[ \text{Volume wasted} = \text{Volume of cone} - \text{Volume of cylinder} \]

\[ \text{Volume wasted} = 1609.14 - 678.86 \approx 930.28 \, \text{cm}^3 \]

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

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

Substituting the values:

\[ \text{Percentage wasted} = \left( \frac{930.28}{1609.14} \right) \times 100 \]

Calculating this gives:

\[ \text{Percentage wasted} \approx 57.8\% \]

### Final Answer
Thus, the percentage of material wasted during the process is approximately **57.8%**.

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