To find the radius of each sphere, we can use the principle of conservation of volume. The volume of the metallic cone is equal to the total volume of the 18 spheres formed.

Step 1: Find the volume of the cone

The formula for the volume of a cone is:

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

Given:

• Base diameter of the cone = 3 cm, so radius $r = \frac{3}{2} = 1.5$ cm
• Height $h = 12$ cm

Now, we can calculate the volume of the cone:

$V_{\text{cone}} = \frac{1}{3} \pi (1.5)^2 (12)$

$V_{\text{cone}} = \frac{1}{3} \pi \times 2.25 \times 12 = 9 \pi \, \text{cm}^3$

Step 2: Find the total volume of the 18 spheres

The formula for the volume of a sphere is:

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

Let the radius of each sphere be $r_{\text{sphere}}$. The total volume of 18 spheres will be:

$18 \times \frac{4}{3} \pi r_{\text{sphere}}^3 = 9 \pi$

Step 3: Solve for the radius of each sphere

Cancel out $\pi$ from both sides of the equation:

$18 \times \frac{4}{3} r_{\text{sphere}}^3 = 9$

$24 r_{\text{sphere}}^3 = 9$

Now, solve for $r_{\text{sphere}}^3$:

$r_{\text{sphere}}^3 = \frac{9}{24} = 0.375$

Finally, take the cube root of both sides to find the radius:

$r_{\text{sphere}} = \sqrt[3]{0.375} \approx 0.72 \, \text{cm}$

Conclusion:

The radius of each sphere is approximately 0.72 cm.

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Would you like any further details or have any other questions?

Here are 5 related questions:

1. How is the volume of a cone derived from its base and height?
2. What is the significance of the cube root in volume calculations?
3. How would the radius change if only 9 spheres were made from the cone?
4. What other shapes could be formed by melting the cone, aside from spheres?
5. How does the volume of a sphere change if the radius is doubled?

Tip: Always check that units match across your calculations to avoid errors.