SOLUTION: A cylinder is inscribed inside a sphere of Radius R. Suppose the height of the cylinder is x. write a formula for the volume V(x) of the cylinder as a function of x.

Question 244866: A cylinder is inscribed inside a sphere of Radius R. Suppose the height of the cylinder is x. write a formula for the volume V(x) of the cylinder as a function of x.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The volume of a cylinder, using "x" for the height and "r" for the radius of the circular bases, is:

The problem is to express r in terms of x and R. (It will not be possible, given the information you provided, to express r just in terms of x.)

The keys to this solution are:

The center of the sphere and the center of the cylinder will be the same point.
The distance from the center of the cylinder to the center of either of the two circular bases will be one half of the height of the cylinder.
The points of the two circular bases of the cylinder will be on the sphere, too.

Once you understand the above, you will understand the following diagram:

A = the center of the sphere
B = the center of one of the circular bases of the cylinder
r = radius of the circular base of the cylinder
R = radius of the sphere
x = height of the cylinder

This is a right triangle so we can use the Pythagorean Theorem:

Simplifying we get:

Solving for r:

(discarding the negative square root because radii are not negative)

Now we can subsitute for r in the Volume formula:

Simplifying we get:

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