I presented this as a problem to my calc class, after introducing volumes by the disk and shell method. I gave each student a sheet of cm graph paper, and a slice of cake:
Students drew the volume of rotation and found the outer and inner radii. It wasn't too challenging for them, since the cake slice was just a rectangle. This one was my attempt to do the problem with them:
To check for accuracy, I filled up the original cake pan with water.
The result: 1700 mL. I calculated 2412.7 mL. I was really hoping for more accurate results. I am not sure what went wrong other than the sides of the pan were a little slanted, and we treated them like they were vertical.
Then, to practice the shell method, we used one of these guys:
Here is one student sketching out the cake.
Students tried to use a parabola to model the shape of the cake, which seemed like a good choice. They knew how to use transformations to flip the parabola and translate it to the right location, but they didn't know how to adjust the width. They ended up choosing a random fraction like 1/2 or 1/3 because they knew a fraction in front of x^2 would make it wider. But the results were no where close to the actual cake volume.
Then again, we were running out of time. It was right before lunch. Everyone was hungry. The room smelled like cake. And everybody wanted to eat cake more than they cared about the exact formula for this parabola, or even if it should have been a parabola.
I like the concept of this activity, but I would like to figure out what adjustments to make to get more accurate results next time. I probably just didn't give my students enough time and/or resources to really figure out the right equation for the cross-section.
In the end, the cake was good. We did some math. And everyone thanked me for having a cake day in calculus. So it wasn't a total disaster.