Volume of Cone with Inscribed Sphere

[zeion]

Homework Statement

A cone is circumscribed around a sphere. The radius of the sphere is 5 units.
Write the volume of the cone in terms of x.
There is a diagram.. I will try to describe it:
It is a cross section of the object (sphere in cone). From the center of the circle to the bottom left vertice of the triangle is length 5 + x. (5 is the radius, x is the rest of the line).

Homework Equations

The Attempt at a Solution

So I can get r of the cone with Pythagoras = sqrt((x + 5)^2 - 25). The height would be 10 + something. I'm not sure how the missing part of the height relates to x.

 

[HallsofIvy]

The problem is that there are an infinite number of such cones. First draw a triangle circumscribing a circle (the sphere inside the cone seen from the side). Choose and angle for the vertex at the top of the sphere. That angle cannot be 0 or 180 degrees but it can be any other between. And then there exist a cone, having that angle at the vertex, circumscribing the sphere.

 

[zeion]

So how can I write the volume in terms of x ?

 

[LCKurtz]

I certainly wouldn't have chosen x as the main variable in this problem, but never mind that. Look at the figure. All you need to find the volume of the cone is its radius and height. So in the figure you need to get r and y in terms of x. r is easy from the right triangle AOB. Then you can get w + v in terms of y and x from triangle ABD. Then use the similarity of triangles ABD and DCO to get y in terms of x.