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 radius of the sphere and the number of cones recast.

Answer

To solve the problem, we need to find the radius of the sphere and the number of cones that can be formed from it after melting. Let's break down the solution step by step.

### Step 1: Calculate the radius of the sphere
The formula for the surface area of a sphere is given by:
\[ \text{Surface Area} = 4\pi r^2 \]
We know the surface area of the sphere is \(2464 \, \text{cm}^2\). Setting up the equation:
\[ 4\pi r^2 = 2464 \]
We can substitute \(\pi\) with \(\frac{22}{7}\) for calculation:
\[ 4 \times \frac{22}{7} \times r^2 = 2464 \]
Now, we can solve for \(r^2\):
\[ r^2 = \frac{2464 \times 7}{4 \times 22} \]
Calculating the right side:
\[ r^2 = \frac{2464 \times 7}{88} = \frac{17248}{88} = 196 \]
Taking the square root to find \(r\):
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Step 2: Calculate the volume of the sphere
The volume \(V\) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]
Substituting \(r = 14 \, \text{cm}\):
\[ V = \frac{4}{3} \times \frac{22}{7} \times (14)^3 \]
Calculating \(14^3\):
\[ 14^3 = 2744 \]
Now substituting this value back into the volume formula:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 2744 \]
Calculating:
\[ V = \frac{4 \times 22 \times 2744}{3 \times 7} = \frac{241792}{21} \approx 11509.619 \, \text{cm}^3 \]

### Step 3: Calculate the volume of one cone
The volume \(V_c\) of a cone is given by:
\[ V_c = \frac{1}{3}\pi r_c^2 h_c \]
Where \(r_c = 3.5 \, \text{cm}\) and \(h_c = 7 \, \text{cm}\):
\[ V_c = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times 7 \]
Calculating \((3.5)^2\):
\[ (3.5)^2 = 12.25 \]
Substituting back:
\[ V_c = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 7 \]
The \(7\) cancels out:
\[ V_c = \frac{22 \times 12.25}{3} = \frac{272.5}{3} \approx 90.833 \, \text{cm}^3 \]

### Step 4: Calculate the number of cones
Let \(n\) be the number of cones formed. Since the volume of the sphere is equal to the total volume of the cones, we have:
\[ n = \frac{V}{V_c} \]
Substituting the values:
\[ n = \frac{11509.619}{90.833} \approx 126.5 \]
Since \(n\) must be a whole number, we round down:
\[ n = 126 \]

### Final Answers
- The radius of the sphere is \(14 \, \text{cm}\).
- The number of cones recast is \(126\).