Some sweet calculus fun

So I was looking for a quick project that would be fun for the Calculus kids to end out the year and came across a couple sites talking about "Bundt Cake Calculus".   Decided to try it and it worked out spectacularly.  I am going to get a bit into the weeds about the calculus, so if that isn't your thing you might want to right now think "mmm, cake" and move on to something else.

For those of you who don't know, a Bundt Cake is a cake baked in a circular pan with a tube in the middle. 

The tube results in more cake being exposed to the heat of baking  The name apparently came from Nordic Ware who started making this aluminum pan in the 50s and trademarked the name "bundt".

So what's the Calculus part?  Well, I started by making a chocolate marble cake in my Bundt pan.  You can see my recipe by following the link to my recipe blog.

We then took a thin slice of cake and laid it on a piece of centimeter graph paper.  Tracing out the slice, we ended up with something that looked like this:





Carefully plotting point around the edge of the tracing we came up with 21 points which we plugged into a TI-84Plus graphing calculator.  A quartic regression (4th power polynomial) gave us the equation:

y = - 0.01941828x^4 + 0.547897x^3 - 5.6421197x^2 +24.86125x - 31.69816607.

The r-squared value (a measure of accuracy of a regression calculation) was 0.942, pretty darn close (1 is perfect, 0 is no relationship between points and equation at all).

We then used the Shells Method to calculate volume of the cake.  For the layman, essentially we took a vertical slice of the graph above, with width delta x, and rotated that slice around the vertical axis.  That creates something that looks like a soup can with the top and bottom removed.  The volume of that can is calculated using the formula 2*pi*x^2 * height * delta x.  So far we haven't even done any calculus, just geometry, but here's where it gets interesting.  You add up all the cans between the two x-intercepts and you have the volume of the cake.  To make it even more accurate, you take the limit as delta x goes to zero.  That equation looks something like this:

 where f(x) is the equation we found above.  So what was the result?

Well, adding water to the Bundt pan with a measuring cup we found that the pan held 2625 cubic centimeters of water (1 ml = 1 cc).

Performing the above calculation we discovered that the cake volume was 2577 cubic centimeters.

Difference (I won't say error because both were fairly approximate) of 48 cc, or about 3 tablespoons.

Slightly less than 2%.  Not to shabby.

And then we all ate cake.