A solid metallic cylinder of height 10 cm and diameter 14 cm is melted to make two cones in the proportion of their volumes as `3:4` , keeping the height 10 cm , what would be the percentage increase in the flat surface area ?

Options

A. `0.09%`
B. `16 %`
C. `50 % `
D. `200 %`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: 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 cylinder
- \( h \) is the height of the cylinder

Given:
- Height \( h = 10 \) cm
- Diameter \( d = 14 \) cm, hence radius \( r = \frac{d}{2} = \frac{14}{2} = 7 \) cm

Substituting the values:
\[ V = \pi (7^2)(10) = \pi (49)(10) = 490\pi \text{ cm}^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx 490 \times \frac{22}{7} = 1540 \text{ cm}^3 \]

### Step 2: Calculate the volumes of the two cones
The cylinder is melted to form two cones in the ratio of their volumes \( 3:4 \). Let the volumes of the two cones be \( 3x \) and \( 4x \).

Since the total volume of the cylinder equals the total volume of the two cones:
\[ 3x + 4x = 1540 \]
\[ 7x = 1540 \implies x = \frac{1540}{7} = 220 \text{ cm}^3 \]

Thus, the volumes of the cones are:
- Volume of Cone 1 \( = 3x = 3 \times 220 = 660 \text{ cm}^3 \)
- Volume of Cone 2 \( = 4x = 4 \times 220 = 880 \text{ cm}^3 \)

### Step 3: Calculate the radius of each cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]

For Cone 1 (Volume = 660 cm³):
\[ 660 = \frac{1}{3} \pi r_1^2 (10) \]
\[ 660 = \frac{10}{3} \pi r_1^2 \implies r_1^2 = \frac{660 \times 3}{10 \pi} = \frac{1980}{10 \pi} = \frac{198}{\pi} \]

For Cone 2 (Volume = 880 cm³):
\[ 880 = \frac{1}{3} \pi r_2^2 (10) \]
\[ 880 = \frac{10}{3} \pi r_2^2 \implies r_2^2 = \frac{880 \times 3}{10 \pi} = \frac{2640}{10 \pi} = \frac{264}{\pi} \]

### Step 4: Calculate the flat surface area of both cones
The flat surface area \( A \) of a cone is given by:
\[ A = \pi r^2 \]

Flat surface area of Cone 1:
\[ A_1 = \pi r_1^2 = \pi \left(\frac{198}{\pi}\right) = 198 \text{ cm}^2 \]

Flat surface area of Cone 2:
\[ A_2 = \pi r_2^2 = \pi \left(\frac{264}{\pi}\right) = 264 \text{ cm}^2 \]

Total flat surface area of both cones:
\[ A_{total} = A_1 + A_2 = 198 + 264 = 462 \text{ cm}^2 \]

### Step 5: Calculate the flat surface area of the cylinder
The flat surface area of the cylinder is given by:
\[ A_{cylinder} = 2 \pi r^2 = 2 \pi (7^2) = 2 \pi (49) = 98\pi \text{ cm}^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A_{cylinder} \approx 98 \times \frac{22}{7} = 308 \text{ cm}^2 \]

### Step 6: Calculate the percentage increase in flat surface area
The percentage increase is calculated as:
\[ \text{Percentage Increase} = \frac{A_{total} - A_{cylinder}}{A_{cylinder}} \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{462 - 308}{308} \times 100 = \frac{154}{308} \times 100 \approx 50\% \]

### Final Answer
The percentage increase in the flat surface area is **50%**.