Application Problems and Soda Cans

Below is a typical 'application' problem from a standard Calculus book.

It's ridiculous.

I believe that there are a number of compelling reasons why applications should be included in a Calculus course: they capture student interest, they encourage problem solving skills, and they demonstrate the power of mathematics. But in all three areas, this example fails. Very few college freshmen are interested in corrals, the pre-labeling of the example with x and y robs the student from any opportunity to develop and exercise problem solving skills, and honestly the solution is unlikely to leave the student with the impression, "Wow, math matters."

Simply put, the application appears artificial and uninteresting. There's no obvious reason why such a problem is preferable to simply giving the student an equivalent problem in strictly geometric terms with no mention of ranchers or corrals.

So what do we do? One approach is to teach 'only the math'; however, I'm concerned this robs the student of the above mentioned benefits, Thus I've settled on a different option.

In Dan Meyer's excellent TED talk, "Math class needs a makeover", he gives some suggestions for presenting application questions. Here is a slightly modified list I've created to guide me in choosing applications:

• Choose a relatable/interesting example. Real-world objects that students are familiar with are especially good choices.
• Form a natural, short question in non-mathematical language.
• Allow the students to formulate this into a more rigorous question.
• Allow students to label and identify the information essential to solving the problem.
• Take time to appreciate the solution and its significance.

Below is one such application I've developed and used in teaching.

Soda Can Problem
Holding a soda can, one can naturally pose the question, "How did Coca Cola decide on the dimensions of this can?" After all, they could have made it taller and skinner or shorter and fatter, but for some reason the standard can size was settled on.

Notice that a soda can holds 12 ounces (355 ml). Thus we're really asking what are the best dimensions so that our volume comes out to be 355 cubic cm. Of course, 'best' means cheapest and cheapest means using the least amount of aluminum possible in the construction of the can.

Now we can formulate our question more precisely: what dimensions of a can (that is, a cylinder) minimizes the material of the can (that is, the surface area) but keep the volume constant at 355 cubic cm.

Diameter and radius arise as natural choices to determine the dimensions of the can, and from these one can derive expressions for surface area and volume.

This is how I've begun my lectures on related rates in single variable calculus and Lagrangian multipliers in vector calculus. We then go on to develop the necessary calculus to solve the problem.

Then comes the best part: at the end of the lecture when we calculate the solution, it is actually interesting.

We calculate what the dimensions should be to minimize surface area, then measure with a ruler the actual can to see if they agree.

They don't.

Coca Cola would be saving a significant amount of material (and hence money) if they made their cans significantly shorter and fatter.

So why don't they? Maybe none of their product engineers know Calculus. More likely, a shorter fatter can size, although cheaper to make, would be awkward to hold. Perhaps they intentionally paid more for greater consumer satisfaction.

How did they model that trade-off? What other factors did they consider?

The student is left with an interesting result but also a host of related questions. Questions that calculus can continue to help them answer.