A fun calculus problem -> folding a circle wedge into a cone

I’m a few days late publishing this exercise – my son finished up the section on applied max / min problems last week. But I thought his work on this problem was fascinating and wanted to publish it even if it was a little late.

So, last week my son came across this max / min problem in his calculus book:

It gave him a little trouble and since I was on the road for work it wasn’t so easy to help him. We went through the problem when I got back from a trip -> I thought it would be fun to start from the beginning and actually make some cones before diving into the problem.

Next we started down the path of trying to work through the problem. Here’s how he got started:

In the last video he was able to write down an expression for the volume of the cone in terms of the angle of the wedge. In this video he writes down a variant of that expression (the square of the volume) and gets ready to find the maximum volume:

Now that he has a relatively simple expression for the volume squared, he finds the derivative to find the angle giving the maximum volume:

Finally – he calculated the maximum volume. The expression for the angle is a little messy, but the maximum volume has a (slightly) easier form.

Overall, I think this is a great problem for kids learning calculus. It also pulls in a little 3d geometry and 2d geometry review, which was nice.

With this section about applied max / min problems done, we are moving on to integration 🙂