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.

Algebra ->  Volume -> 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.      Log On


   

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) About Me  (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:
V+=+pi%2Ar%5E2%2Ax
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:
%28x%2F2%29%5E2+%2B+r%5E2+=+R%5E2
Simplifying we get:
x%5E2%2F4+%2B+r%5E2+=+R%5E2
Solving for r:
r%5E2+=+R%5E2+-+x%5E2%2F4
r+=+sqrt%28R%5E2+-+x%5E2%2F4%29 (discarding the negative square root because radii are not negative)

Now we can subsitute for r in the Volume formula:
V+=+pi%2A%28sqrt%28R%5E2+-+x%5E2%2F4%29%29%5E2%2Ax
Simplifying we get:
V+=+pi%2A%28R%5E2+-+x%5E2%2F4%29%2Ax
V+=+pi%2Ax%2AR%5E2+-+pi%2Ax%5E3%2F4