EXTENSION: A sphere of radius r is inscribed in a cone with base radius R and height h. Find the relationship between r, R, and h. Additionally, generalize your solution to n

Step 1: Visualize the cross-section

To analyze the relationship, we consider a 2D cross-section of the cone and the inscribed sphere. This cross-section forms an isosceles triangle (representing the cone) with a circle inscribed within it (representing the sphere). The height of the cone is $h$, and its base radius is $R$. The radius of the inscribed sphere is $r$. We can identify two similar right-angled triangles in this cross-section.

Step 2: Identify similar triangles and their properties

Let the cone's vertex be A, the center of its base be D, and a point on the base circumference be B. The center of the inscribed sphere is O, and its radius is $r$. Let E be the point where the sphere touches the slant height AB. Triangle $ADE$ is a right-angled triangle with height $h$ and base $R$. Triangle $AEO$ is also a right-angled triangle, where $OE = r$ and $AO = h - r$. The triangles $\triangle ADE$ and $\triangle AEO$ are similar because they share angle $DAE$ and both have a right angle.

Step 3: Set up the proportion from similar triangles

From the similarity of $\triangle AEO$ and $\triangle ADB$ (where $DB = R$ and $AB = \sqrt{R^2 + h^2}$ is the slant height), we can write the ratio of corresponding sides. The ratio of the radius of the sphere to the base radius of the cone is equal to the ratio of the distance from the vertex to the sphere's center to the slant height of the cone.

Step 4: Solve for r

Now, we cross-multiply the terms in the proportion to eliminate the denominators. This step helps us to isolate $r$ and express it in terms of $R$ and $h$.

Step 5: Rearrange the equation to find the relationship

Expand the equation and gather all terms containing $r$ on one side. Factor out $r$ and then divide to solve for $r$. This gives us the relationship between the sphere's radius $r$, the cone's base radius $R$, and its height $h$.

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