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 radius of the solid sphere, and the no. of cones cast (π=3.14).

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the radius of the solid sphere

The formula for the surface area \( A_s \) of a sphere is given by:

\[ A_s = 4 \pi r^2 \]

We know the total surface area \( A_s = 1256 \, \text{cm}^2 \). We can rearrange the formula to solve for \( r \):

\[ r^2 = \frac{A_s}{4 \pi} \]

Substituting the values:

\[ r^2 = \frac{1256}{4 \times 3.14} \]

Calculating the denominator:

\[ 4 \times 3.14 = 12.56 \]

Now substituting back:

\[ r^2 = \frac{1256}{12.56} \approx 100 \]

Taking the square root to find \( r \):

\[ r = \sqrt{100} = 10 \, \text{cm} \]

### Step 2: Calculate the volume of the solid sphere

The volume \( V_s \) of the sphere is given by:

\[ V_s = \frac{4}{3} \pi r^3 \]

Substituting \( r = 10 \, \text{cm} \):

\[ V_s = \frac{4}{3} \times 3.14 \times (10)^3 \]

Calculating \( (10)^3 \):

\[ (10)^3 = 1000 \]

Now substituting back:

\[ V_s = \frac{4}{3} \times 3.14 \times 1000 \]

Calculating \( \frac{4 \times 3.14 \times 1000}{3} \):

\[ V_s \approx \frac{12560}{3} \approx 4186.67 \, \text{cm}^3 \]

### 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 \( r_c = 2.5 \, \text{cm} \) and \( h_c = 8 \, \text{cm} \):

\[ V_c = \frac{1}{3} \times 3.14 \times (2.5)^2 \times 8 \]

Calculating \( (2.5)^2 \):

\[ (2.5)^2 = 6.25 \]

Now substituting back:

\[ V_c = \frac{1}{3} \times 3.14 \times 6.25 \times 8 \]

Calculating \( 6.25 \times 8 \):

\[ 6.25 \times 8 = 50 \]

Now substituting back:

\[ V_c = \frac{1}{3} \times 3.14 \times 50 \approx \frac{157}{3} \approx 52.33 \, \text{cm}^3 \]

### Step 4: Calculate the number of cones that can be cast

To find the number of cones \( n \) that can be cast from the melted sphere, we use the formula:

\[ n = \frac{V_s}{V_c} \]

Substituting the volumes we calculated:

\[ n = \frac{4186.67}{52.33} \approx 80 \]

### Final Answers

- The radius of the solid sphere is \( 10 \, \text{cm} \).
- The number of cones cast is \( 80 \).

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