A solid sphere of radius 15 cm 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 of finding the number of cones recast from a melted solid sphere, we will follow these steps:

### Step 1: Calculate the Volume of the Sphere
The formula for the volume \( V_s \) of a sphere is given by:
\[ V_s = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

Given:
- Radius of the sphere \( R = 15 \, \text{cm} \)

Substituting the value into the formula:
\[ V_s = \frac{4}{3} \pi (15)^3 \]

Calculating \( 15^3 \):
\[ 15^3 = 3375 \]
Thus,
\[ V_s = \frac{4}{3} \pi (3375) = 4500 \pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One Cone
The formula for the volume \( V_c \) of a cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

Given:
- Radius of the cone \( r = 2.5 \, \text{cm} \)
- Height of the cone \( h = 8 \, \text{cm} \)

Substituting the values into the formula:
\[ V_c = \frac{1}{3} \pi (2.5)^2 (8) \]

Calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Thus,
\[ V_c = \frac{1}{3} \pi (6.25)(8) = \frac{1}{3} \pi (50) = \frac{50}{3} \pi \, \text{cm}^3 \]

### Step 3: Calculate the Number of Cones
To find the number of cones \( n \) recast from the melted sphere, we use the relationship:
\[ n = \frac{V_s}{V_c} \]

Substituting the volumes we calculated:
\[ n = \frac{4500 \pi}{\frac{50}{3} \pi} \]

The \( \pi \) cancels out:
\[ n = \frac{4500}{\frac{50}{3}} = 4500 \times \frac{3}{50} = \frac{13500}{50} = 270 \]

### Final Answer
The number of cones recast is \( \boxed{270} \).

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