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.
Hi Amy. Great post and I can't wait to try it in class. I experimented with my wife's bunt pan and found that the slopes really do make a difference. For my pan (which is 10 cm high), I found the radius of the inner circle changed by .4 cm and the outer circle by nearly 1 cm. The disc method then gave me a volume (about 4170 cm^3) which was less than 50 cm^3 off of the result I got with your water method.ReplyDelete
For students it would be great to experiment first with the calculation where you don't worry about the slope, see the error, and then the faster students could work out the volume with the slope included.
GLSR, Thank you so much for trying this and for sharing your results! I like the idea of doing one calculation without slope, then considering the slope for a more accurate result. Maybe we will do this again before the year is over. No one ever complains when I bring food to class. :) AmyReplyDelete
Liked the post! Very interesting example!
I also think that, in addition to calculating integrals with specified CONVENIENT functions, it might be useful for students to be aware that there are many more arbitrary shapes to calculate the surface and volume of. As a matter of fact, if students look around there are way more shapes and volumes that can not be represented by analytical functions (in closed form). Here is one of my posts how to approach calculus and taking to the limit procedure (as the fundamental idea). You may find it illustrative: "How to Calculate Surface Area of an Arbitrary Shape - Story of Pirate Island" http://goo.gl/hbVEY
Nesha Carad, P. Eng.