Maximizing Volume: Understanding the Relationship Between a Cone and Sphere

[hotvette]

I was helping my 17 year old daughter (just starting calculus) with the optimization problem of maximizing the volume of a right circular cone that can inscribed in a sphere. She tried what she thought was a short cut by using a cone with vertex at the center the sphere (instead of the top) and couldn’t understand why it didn’t yield the right answer. I tried to explain that she solved a different problem but she couldn’t understand why the solution to the simpler problem wasn’t also the solution to the stated problem. It didn’t help that the very next problem was a distance problem where the book suggested a short cut of minimizing the square of the distance rather than the distance (to avoid square roots). To her, both were the same (i.e. a logical shortcut). I’ve since been struggling with how to explain the apparent discrepancy. I’ve thought about using a triangle/circle analogy and say that the two triangles aren’t similar. Any other ideas?

 

[hgfalling]

A cone with vertex at the center of the sphere wouldn't be inscribed. Comparing a picture of that with a picture of an inscribed cone would seem to show the difference pretty clearly.

 

[hotvette]

A cone with vertex at the center of the sphere wouldn't be inscribed.

Yep, that's clear and is not being debated. What she can't understand (and I've been unable to explain) is why the optimal radius of the smaller problem isn't the same optimal radius of the larger problem. Of course they are two different problems, but I'm looking for more of an explanation than that. Maybe there isn't one.

 

[Office_Shredder]

Maybe it would help to find out why she thinks they would be the same

 

[hotvette]

Maybe it would help to find out why she thinks they would be the same

Maybe, if I can catch her in a 'low frustration' and 'rational' mood. Not easy with a teenager. The kick in the pants was the next problem (minimize distance from a fixed point to a function) that specifically recommended solving a related but simpler problem instead (i.e. distance squared). The two situations seem the same to her (related and simpler).

Trying hard to keep up her enthusiasm (since she is taking calc B/C in the fall) by offering a thoughtful response.....

 

[Dick]

You could try explaining that dr^2/dt=2*r*dr/dt. So if dr^2/dt is zero and r is not zero then dr/dt is also zero. But without a 'low frustration' and 'rational' mood, I doubt that will help.