The surface area of a solid metallic sphere is `2464 cm^(2).` It is melted and recast into solid right circular cones of radius 3.5 cm and height 7 cm. Calculate :
the number of cones recast. (Take `pi = (22)/(7))`

Answer

To solve the problem of finding the number of cones that can be recast from a solid metallic sphere with a given surface area, we will follow these steps:

### Step 1: Calculate the radius of the sphere
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]
Given that the surface area \( S = 2464 \, \text{cm}^2 \) and \( \pi = \frac{22}{7} \), we can set up the equation:
\[ 2464 = 4 \times \frac{22}{7} \times r^2 \]

### Step 2: Rearranging the equation to find \( r^2 \)
First, we will multiply both sides by \( \frac{7}{22} \):
\[ r^2 = \frac{2464 \times 7}{4 \times 22} \]

### Step 3: Simplifying the equation
Calculating the right-hand side:
\[ r^2 = \frac{2464 \times 7}{88} \]
Now, simplify \( 2464 \div 88 \):
\[ 2464 \div 88 = 28 \]
Thus,
\[ r^2 = 28 \times 7 = 196 \]

### Step 4: Finding the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Step 5: Calculate the volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r = 14 \, \text{cm} \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times (14)^3 \]

### Step 6: Calculating \( 14^3 \)
Calculating \( 14^3 \):
\[ 14^3 = 2744 \]
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 2744 \]

### Step 7: Simplifying the volume calculation
Calculating \( \frac{4 \times 22 \times 2744}{3 \times 7} \):
\[ V = \frac{241792}{21} \]

### Step 8: Calculate the volume of one cone
The volume \( V_c \) of a cone is given by the formula:
\[ V_c = \frac{1}{3} \pi r_c^2 h_c \]
Where \( r_c = 3.5 \, \text{cm} \) and \( h_c = 7 \, \text{cm} \):
\[ V_c = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times 7 \]

### Step 9: Calculate \( (3.5)^2 \)
Calculating \( (3.5)^2 = 12.25 \):
\[ V_c = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 7 \]
The \( 7 \) cancels out:
\[ V_c = \frac{1}{3} \times 22 \times 12.25 \]

### Step 10: Final calculation for volume of one cone
Calculating \( 22 \times 12.25 = 270.5 \):
\[ V_c = \frac{270.5}{3} = 90.1667 \, \text{cm}^3 \]

### Step 11: Calculate the number of cones
The number of cones \( n \) that can be made is given by:
\[ n = \frac{V}{V_c} \]
Substituting the volumes:
\[ n = \frac{241792/21}{90.1667} \]

### Step 12: Final calculation for number of cones
Calculating \( n \):
\[ n = \frac{241792}{21 \times 90.1667} \approx 128 \]

Thus, the number of cones that can be recast from the melted sphere is **128**.