Optimization Question: Cylinder inscribed in sphere

[peblez]

Find the dimensions of the right-circular cylinder of greatest volume which can be inscribed in a sphere with a radius of 10 cm.


I really have no idea how to attack this problem, i know the formula for cyl is v = pi r^2 h, but i dont' know how to apply this formula, and i don't know how to maximize this.

can someone show me how

thank you

 

[morson]

Key word: "right." Imagine a line going from the bottom part where the cylinder touches the
sphere to the top part where the cylinder touches the sphere on the opposite side.

By Pythagoras, that length, d, is equal to: \(\displaystyle \L\sqrt{x^2 + h^2}\), where x is the
diameter of the cylinder and h is the height of the cylinder.

We know that d is also the diameter of the sphere. So, d = 20.

Therefore: \(\displaystyle \L\ x^2 + h^2 = 400 [1]\)

The volume, as you pointed out, of the cylinder, is \(\displaystyle \L\ V = \pi\ r^2 h\)

Now, x = 2r, so r = \(\displaystyle \frac{x}{2}\\), therefore: \(\displaystyle \L\ V = \pi\ \frac{x^2}{4}\ h [2]\)

Solving for \(\displaystyle x^2\) in [1]: \(\displaystyle \L\ x^2 = 400 - h^2\)

Substituting in [2], we have:

\(\displaystyle \L\ V = \frac{\pi\(400 - h^2)h}{4}\\)

So what value of h will maximize V?