# Optimization of a Cylinder In a Sphere WITHOUT Using Calculus

I have a quick question. I'm curious as to how to do an optimization question WITHOUT using calculus.

Question: Determine the dimensions of the cylinder of maximum volume that can be inscribed in a sphere with radius 8cm.

I know how to do the question using calculus, but my sister asked me it and I don't know how to teach her the question without calculus.

I know if I was using single-variable calculus, I would use Pythagorean Theorem and isolate for one variable, then take that equation and put it into the equation for volume and determine the first derivative for the equation of volume and set the equation to 0 to determine the critical points. But my sister won't have any idea about what I'm talking about if I tell her that, so I'm just stuck as to how I would answer this without calculus!

• We just have to use the inequality $a+a+b\geq 3(a^2 b)^{1/3}$; it is not difficult to prove it with elementary techniques. Commented Nov 20, 2014 at 23:54
The maximum of $V=2\pi r^2 d$ under the constraint $r^2+d^2=64$ is attained when $r^2=2d^2$, since: $$64 = \frac{1}{2}r^2 + \frac{1}{2}r^2 + d^2 \geq 3\left(\frac{1}{4}r^4d^2\right)^{1/3} = 3\left(\frac{1}{2}r^2 d\right)^{2/3}$$ by the AM-GM inequality, so: $$r^2d \leq 2\left(\frac{64}{3}\right)^{3/2}$$ and $$V = 2\pi r^2 d \leq 4\pi\cdot\left(\frac{64}{3}\right)^{3/2}=\frac{1024\,\pi}{3\sqrt 3}.$$