# 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!

• You could use this as a motivation to learn calculus either now or in the future (depending on the math level your sister is at). Often even in more advanced math books if something is really easy to do with some extra theory it is stated and the fact that this extra theory makes it easy (or possible) to do this is stated with the idea being it gives the reader the idea of why to study this new topic.
– user171177
Nov 20, 2014 at 23:04
• Use the AM-GM inequality. Nov 20, 2014 at 23:07
• But my sister won't be learning the AM-GM inequality in grade 9 :( Nov 20, 2014 at 23:35
• 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. 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 $$V = 2\pi r^2 d \leq 4\pi\cdot\left(\frac{64}{3}\right)^{3/2}=\frac{1024\,\pi}{3\sqrt 3}.$$