To find the number of small spheres formed from a melted solid cone, we need to equate the volume of the cone to the total volume of the small spheres. Let's break down the process step-by-step:

1. Volume of the Cone:

The formula for the volume of a cone is:

$V_{\text{cone}} = \frac{1}{3} \pi r^2 h$

where:

• $r = 5 \, \text{cm}$ is the radius of the cone,
• $h = 8 \, \text{cm}$ is the height of the cone.

Substituting these values:

$V_{\text{cone}} = \frac{1}{3} \pi (5)^2 (8) = \frac{1}{3} \pi (25) (8) = \frac{1}{3} \pi (200) = \frac{200\pi}{3} \, \text{cm}^3$

2. Volume of One Sphere:

The formula for the volume of a sphere is:

$V_{\text{sphere}} = \frac{4}{3} \pi r^3$

where $r = 0.5 \, \text{cm}$ is the radius of the small sphere. Substituting this value:

$V_{\text{sphere}} = \frac{4}{3} \pi (0.5)^3 = \frac{4}{3} \pi \times 0.125 = \frac{4\pi}{24} = \frac{\pi}{6} \, \text{cm}^3$

3. Number of Spheres:

To find the number of spheres, divide the volume of the cone by the volume of one sphere:

$\text{Number of spheres} = \frac{V_{\text{cone}}}{V_{\text{sphere}}} = \frac{\frac{200\pi}{3}}{\frac{\pi}{6}} = \frac{200\pi}{3} \times \frac{6}{\pi} = \frac{200 \times 6}{3} = 400$

Final Answer:

The number of spheres formed is $400$.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the number of spheres change if the radius of each sphere were doubled?
2. What is the total surface area of all the spheres formed?
3. How would you calculate the volume of a cylinder instead of a cone?
4. What is the relationship between the volume of a cone and its surface area?
5. How does the number of spheres change if the cone has a different height?

Tip: Whenever dealing with volumes, remember that dimensions like radius and height are cubed or squared, which makes the units very important!