smokerblog

Ice Cream Math

Okay. So while I was getting myself some ice cream tonight (Edy's Vanilla Bean) I thought about my dad's peculiar ice-cream-scooping methodology. He likes to scoop ice cream from the half-gallon boxes (usually Perry's or Breyer's) in rectangular shapes, the goal being to leave as little surface area as possible exposed for freezer burn. I've always assumed, but never proved, that this is, in fact, the best way to go about achieving that goal. Tonight, I decided to find out for sure.

First, let's assume that the carton of ice cream measures 8 x 4 x 4 inches, like so:

Now, I figure that there are eight servings in each carton--there are probably more, but knowing how much my dad likes ice cream, let's stick with eight. And let's assume that he eats one serving every night. The first serving would leave 16 square inches of ice cream exposed, thusly:

The next night, the second serving would also leave 16 square inches exposed:

Hmmm, except 8 of those square inches were already exposed the night before, so let's suppose that the effects are cumulative and we can call this 24 total inches exposed.

Okay, now the typical ice cream consumer would just start scooping from the middle leaving a roughly semi-spherical hole of exposed ice cream, (use your imagination):

And here comes the geeky math part. (Matt had better check my work.) We know that each day, dad consumes 16 cubic inches of ice cream (2 x 2 x 4). To find the total exposed area of the semi-spherical indentation, we will use the volume of the ice cream scooped to find the radius of the indentation. Then we can use the radius to determine the surface area.

The formula for the volume of our half-sphere shaped hole:

The formula for the surface area of the hole:

Solving for r in the first equation and plugging it in to the second equation results in:

(okay, I'm busted, this was all really just an excuse to play with LaTeX)

Solving this equation gives a value in the neighborhood of 24.4 square inches, more than Dad's method even considering the cumulative effect. Turns out, the next night, as the hole gets deeper, the exposed surface area is 38.7 square inches. Freezer burn city. Dad's right...I never doubted it.