The total area of a solid metallic sphere is `1256 cm^(2).` It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate :
the number of cones recast.

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the radius of the sphere
The total surface area of a sphere is given by the formula:
\[ \text{Total Surface Area} = 4\pi r^2 \]
Given that the total surface area of the sphere is \(1256 \, \text{cm}^2\), we can set up the equation:
\[ 4\pi r^2 = 1256 \]
Substituting \(\pi \approx 3.14\):
\[ 4 \times 3.14 \times r^2 = 1256 \]
This simplifies to:
\[ 12.56 r^2 = 1256 \]
Now, divide both sides by \(12.56\):
\[ r^2 = \frac{1256}{12.56} = 100 \]
Taking the square root gives:
\[ r = \sqrt{100} = 10 \, \text{cm} \]

### Step 2: 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 \(r = 10 \, \text{cm}\):
\[ V = \frac{4}{3} \times 3.14 \times (10)^3 \]
Calculating \(10^3 = 1000\):
\[ V = \frac{4}{3} \times 3.14 \times 1000 \]
Calculating further:
\[ V = \frac{12560}{3} \approx 4186.67 \, \text{cm}^3 \]

### Step 3: Calculate the volume of one cone
The volume \(V\) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Given the radius \(r = 2.5 \, \text{cm}\) and height \(h = 8 \, \text{cm}\):
\[ V = \frac{1}{3} \times 3.14 \times (2.5)^2 \times 8 \]
Calculating \((2.5)^2 = 6.25\):
\[ V = \frac{1}{3} \times 3.14 \times 6.25 \times 8 \]
Calculating \(6.25 \times 8 = 50\):
\[ V = \frac{1}{3} \times 3.14 \times 50 \]
Calculating further:
\[ V = \frac{157}{3} \approx 52.33 \, \text{cm}^3 \]

### Step 4: Calculate the number of cones recast
To find the number of cones that can be made from the volume of the sphere, we divide the volume of the sphere by the volume of one cone:
\[ \text{Number of cones} = \frac{\text{Volume of sphere}}{\text{Volume of one cone}} = \frac{4186.67}{52.33} \approx 80 \]

### Final Answer
The number of cones recast is approximately \(80\).

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