A sphere is inscribed in a cone of radius 5 and height 12. Find the radius of the sphere

To solve the problem of finding the radius of the sphere inscribed in the cone, let's denote the following:

- \( r \) is the radius of the inscribed sphere.
- \( R = 5 \) is the radius of the base of the cone.
- \( h = 12 \) is the height of the cone.
- \( H \) is the height of the cone from the vertex to the center of the sphere.

### Step 1: Use the relation between the dimensions
For a sphere inscribed in a right circular cone, the radius \( r \) of the sphere is given by the formula:
\[
r = \frac{R \times (h - H)}{\sqrt{R^2 + h^2}}
\]
where \( H \) is the distance from the vertex of the cone to the center of the sphere. \( H \) can be derived from:
\[
H = \frac{h^2 \times R}{R^2 + h^2}
\]

### Step 2: Calculate \( H \)
Let's first calculate \( H \):
\[
H = \frac{12^2 \times 5}{5^2 + 12^2} = \frac{144 \times 5}{25 + 144} = \frac{720}{169} \approx 4.26
\]

### Step 3: Calculate \( r \)
Now that we have \( H \), let's substitute into the formula for \( r \):
\[
r = \frac{5 \times (12 - 4.26)}{\sqrt{5^2 + 12^2}} = \frac{5 \times 7.74}{\sqrt{25 + 144}} = \frac{38.7}{\sqrt{169}} = \frac{38.7}{13} \approx 2.98
\]

So, the radius of the inscribed sphere is approximately \( r \approx 2.98 \) units.

Would you like more details on any part of the calculation?

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Here are five related questions to explore further:

1. How would the radius of the sphere change if the cone's height were doubled?
2. What is the volume of the sphere inscribed in the cone?
3. If the cone had a different base radius, how would that affect the radius of the inscribed sphere?
4. How do you derive the formula for the radius of a sphere inscribed in a cone?
5. What is the surface area of the sphere inscribed in the cone?

**Tip:** Understanding the geometry of conic sections can help solve more complex problems involving spheres and cones.

Learn how to find the radius of a sphere inscribed in a cone with a base radius of 5 units and height 12 units. Detailed step-by-step solution involving geometry of cones and spheres. Suitable for high school students studying geometry and calculus.

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