# Cylinder in a sphere

1. Nov 21, 2007

### skateza

1. The problem statement, all variables and given/known data
Find the volume of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10cm.

I'm using this problem to help me solve a similar one with a cylinder inside a cone, now what i'm not sure about is, in the answer book they say, Let the radius of the cylener be r cm, 0 < r< 10. Then the height is 2sqrt(100-r^2)
.... where did they get this height from?

2. Nov 21, 2007

### HallsofIvy

Staff Emeritus
Draw a picture. "Seen from the side", the sphere is a circle with radius 10. Now draw a "cylinder" (i.e. a rectangle) in the "sphere" (circle). If the radius of the cylinder is r, then the base length of the rectangle is 2r. Let h be the height of the cylinder (rectangle) and draw a diagonal. What is the length of the diagonal? Can you use the Pythagorean theorem to write h as a function of r?

3. Nov 21, 2007

### skateza

okay with that i still can't figure out my peoblem. Here is the question i am really trying to solve. A right cirular cylinder is inscribed in a cone with height 3m, and base radius 3m. Find the largest possible volume of such a cylinder.

V = (pie)r^2h, how would i find the height in this casE?

4. Nov 21, 2007

### skateza

Okay i think i got it, is this right:

Drawing a side diagram with a triangle and a rectangle in the middle i can use similar triangles to show cos(Theta) = h/(3-r) = 1; therefore h = 3-r

Using this i get a maximum value of 4pie

5. Nov 21, 2007

Looks right!