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Recently, a quick poll of our students indicated that, contrary to some present fashions in calculus reform, neither epidemiology or population dynamics was on our 18-year-olds' short list of preoccupations. We did find that they thought a lot about food, especially fast food, so we decided to post a fast-food problem. Hamburgers and pizzas being geometrically trivial, we settled on the problem of finding the volume of a taco. A taco is the solid formed by bending a circular tortilla partway around a cylinder and filling it in the obvious way - to the border, but not beyond! A natural problem is to find shapes of cylinders that yield tacos of large volume. Better still, we sought the shape that yields the biggest taco - the taco of largest possible volume for a given tortilla. Unfortunately, we soon found that this was too tall an order for our calculus students to fill; in fact, it is a nontrivial problem in the calculus of variations. But with the help of a computer algebra system, students at all levels can grab hold of this problem. We used Mathematica to plot graphs, evaluate integrals, approximately solve transcendental equations, and search for the extreme values of functions. The main lesson our students carried away is that while many problems cannot be solved exactly in terms of standard functions, reasonable approximations can often be found by a combination of mathematical savvy and computational power. The taco problem offers not only an attractive entry into the calculus of variations, but also a painless encounter with several special functions that students traditionally meet in more complicated circumstances.
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