A solid cone of radius 5 cm and height 8 cm is melted and made into small spheres of radius 0.5 cm. Find the number of spheres formed.

Answer

To solve the problem of finding the number of small spheres formed from a solid cone, we will follow these steps:

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

Given:
- Radius of the cone \( r = 5 \) cm
- Height of the cone \( h = 8 \) cm

Substituting these values into the formula:
\[ V = \frac{1}{3} \pi (5)^2 (8) \]
\[ V = \frac{1}{3} \pi (25)(8) \]
\[ V = \frac{1}{3} \pi (200) \]
\[ V = \frac{200}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one small sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given:
- Radius of the sphere \( r = 0.5 \) cm

Substituting this value into the formula:
\[ V = \frac{4}{3} \pi (0.5)^3 \]
Calculating \( (0.5)^3 \):
\[ (0.5)^3 = \frac{1}{8} \]
Thus:
\[ V = \frac{4}{3} \pi \left(\frac{1}{8}\right) \]
\[ V = \frac{4}{24} \pi \]
\[ V = \frac{1}{6} \pi \text{ cm}^3 \]

### Step 3: Find the number of spheres formed.
Let \( n \) be the number of spheres formed. Since the volume of the cone is equal to the total volume of the spheres, we can set up the equation:
\[ \text{Volume of cone} = n \times \text{Volume of one sphere} \]
Substituting the volumes we calculated:
\[ \frac{200}{3} \pi = n \times \frac{1}{6} \pi \]

### Step 4: Simplify the equation.
Dividing both sides by \( \pi \):
\[ \frac{200}{3} = n \times \frac{1}{6} \]
Multiplying both sides by 6 to solve for \( n \):
\[ 6 \times \frac{200}{3} = n \]
\[ n = \frac{1200}{3} \]
\[ n = 400 \]

### Conclusion:
The number of small spheres formed is \( \boxed{400} \).

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