Well, since you asked...

We've been looking at the volume of prisms, cylinders and cones this week.  I had students working on a project where they had to build one of each with equal heights and widths/diameters.  The idea is to explore the volume of each and see how the eventual formulas will relate to one another.

Then, Jacob traces can on his paper and cuts out the circle.  He cuts a radius and begins rolling the paper (as if he's cut out different sized sectors) to make different cones.  He comes up and says, "Mr. Cox,  I think the cone that is almost flat has the highest volume because the tighter I roll the paper, the less stuff I can fit in it."

Me:  What if the circle is flat?  What's the volume then?

J: There isn't any volume.

Me:  So then when does the cone go from 'flat' to having the most possible volume?

J:  ...

Me: ...

J: What do you mean?

Me:  Maybe there's some kind of sweet spot where the volume gets bigger then starts to get smaller.

J:  Let me think about that.

At this point, I was with Jacob.  I didn't really know what the volume did as the cone changed.  But we were both interested.

The next day, Jacob comes in and says, "Mr. Cox, I thought about what you were saying and I think you're right, there has to be some kind of sweet spot."

So, we sit down and go to work.

I'm thinking about how to model this thing and Jacob enlisted the help of a friend to gather data.  They're cutting sectors from a circle and making cones.  Jacob has dibs on 30, 60, 90, ... degrees and Armando has 15, 45, 75, ...

Our first bit of trouble came when Jacob said, "I can find the radius of the cone's base, but I'm having trouble getting the height because of this..."

Wish these rulers came with a bubble level. 

But, we figured out that the Pythagorean Theorem was a nice work around.

Now, does our data match the model?

It took a while, and thanks to CalcDave for cleaning things up, but this is a pretty cool function.

 Desmos graph is here.


We're estimating the maximum to be about 66 degrees.  And because my calculus is a little rusty, I'm thankful for the folks at WolframAlpha.

This particular function is using a circle with radius = 3.1. 

The function is a little dense at this point, but Jacob was dialed in as we talked about it.  The idea that these crazy expressions really just amounted to V_cone = ⅓πr²h blew him away.