To find the radius of the sphere inscribed in a cone, we need to use geometric relationships between the dimensions of the cone and the inscribed sphere. Let the radius of the cone’s base be $R = 5$, the height of the cone be $h = 12$, and the radius of the inscribed sphere be $r$.

We can use the following formula for the radius of a sphere inscribed in a cone:

$r = \frac{R \cdot h}{\sqrt{R^2 + h^2} + R}$

Step-by-Step Solution:

1. Substitute the given values: $R = 5, \quad h = 12$

2. Calculate $R^2 + h^2$: $R^2 + h^2 = 5^2 + 12^2 = 25 + 144 = 169$

3. Take the square root of $R^2 + h^2$: $\sqrt{169} = 13$

4. Substitute into the formula: $r = \frac{5 \cdot 12}{13 + 5} = \frac{60}{18} = \frac{10}{3}$

Thus, the radius of the inscribed sphere is:

$r = \frac{10}{3} \approx 3.33 \, \text{units}.$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you derive the formula for the radius of an inscribed sphere?
2. What is the volume of the cone in this problem?
3. How does the radius of an inscribed sphere change with the cone’s height?
4. Can the inscribed sphere's volume be calculated using its radius?
5. How would the problem change if the cone had a slant height instead of a vertical height?

Tip: When solving geometric problems with inscribed shapes, understanding the relationship between the dimensions of the larger figure and the inscribed shape can help simplify the problem.