# Thread: Find maximum size of Cylinder that fits in sphere

1. ## Find maximum size of Cylinder that fits in sphere

I'm kinda stuck on this.

I need to find the maximum size of a Cylinder that fits in sphere of radius 1.

Ive come up with the following, .. am I on the right track so far?

Volume of Cylinder = Pi(r^2)h = Pi(r^2)(sqrt(1^2 - r^2))
Volume of Sphere = 4/3(Pi)(r^3)

Pi(r^2)(sqrt(1^2 - r^2)) = 4/3(Pi)(r^3)

Now I just need to solve for r. Is this correct? Where does the differentiation come into it?

2. Originally Posted by floater
I'm kinda stuck on this.

I need to find the maximum size of a Cylinder that fits in sphere of radius 1.

Ive come up with the following, .. am I on the right track so far?

Volume of Cylinder = Pi(r^2)h = Pi(r^2)(sqrt(1^2 - r^2))
Volume of Sphere = 4/3(Pi)(r^3)
You are good up to here.

Pi(r^2)(sqrt(1^2 - r^2)) = 4/3(Pi)(r^3)
No, the volume of the cylinder is not equal to the volume of a sphere. You have already used "fits inside a sphere of radius 1" to deduce that $\displaystyle h= \sqrt{1- r^2}$. Differentiate $\displaystyle \pi r^2(1- r^2)^{1/2}$ and set the derivative equal to 0 to find the r that gives maximum volume.

Now I just need to solve for r. Is this correct? Where does the differentiation come into it?

3. $\displaystyle \pi r^2(1-r^2)^{1/2}$

Well i make the derivitive of the above to be :

$\displaystyle \pi2r(1-r^2)^{1/2} + r^2 1/2(1-r^2)^{-1/2}2r$

I dont know how i would factor that out. Could you give me some hints?

4. Am i right in thinking that i have to factor it first? Into something like

()() = 0

?