Here is a fun optimization problem if anyone would like a whirl.

Find the ELLIPSOID of max volume that can be inscribed in a cone of radius R and height H

No takers, huh? It ain't that bad.

Use similar triangles as you would in any types of these things.

The volume of an ellipse of revolution is V=\frac{4}{3}{\pi}a^{2}b

Where, as you know, a and b are the major and minor axes.

I'd love to take a try, you'll have to guide me though. Ok, what's the question asking really? A cone, in which an ellipse in 3D has to be scribed?

Yes, that's it. But don't over complicate it. These types of problems are all '3D'. Even if you

have, say, a sphere in a cone or a cone is a sphere or what not.


Using similar triangles, we get \frac{a}{H-b}=\frac{R}{H}\Rightarrow b=\frac{H}{R}(R-a)

Oh gosh, now I've got more variables in the volume formula...

V= \frac{4}{3}\pi a^2(\frac{H}{R}(R-a))

Ok I found dV/da, solved to zero and got the optimum volume when

a=\frac{2}{3}R

Yep. That's it. Good deal. And b=\frac{H}{3}

Ah, yup. Didn't know if b was required...

b=\frac{H}{R}(R-a) = \frac{H}{R}(R-\frac{2}{3}R) = \frac{HR}{3R} = \frac{H}{3}

Then, I guess max volume will be replacing the values of a and b?

V=\frac{4}{3} \pi (\frac{2}{3}R)^2 (\frac{H}{3}) = \frac{16\pi R^2H}{81}

See. I told ya' it wasn't that bad. Just a twist on the typical 'cone in sphere', 'sphere in a

Cylinder' type problems. Can you think of another? Maybe a tetrahedron in a sphere or a

Sphere in a tetrahedron?:).

Hey, how about this one:

What is the hyperboloid of max volume that can be inscribed inside a sphere of radius R? :confused:

Just made that up.

You mean a one sheet hyperboloid, or the two sheet one? Anyway... can that be done? :p

Just made a search, the equation for the hyperboloid is

x^2 - y^2 = k

I guess the volume is

[Volume of cylinder radius r] - 2[Integral of curve from center to r] - 2[Curved volume of top or bottom surfaces]