Crops and calculus
Let’s say your neighbor, an apple farmer, decided one day to retire — and, in a magnanimous, fruit-bearing gesture, gave you his orchard. You, naturally, want to make the most of this business and get as many apples from your annual harvest as possible.
Here’s what you know: Each additional tree you plant produces a certain number of additional apples per season. But each additional tree also partially strips the soil of its nutrients, limiting productiveness. So, each apple tree you plant increases your yield ... to a point.
The question is: Where does that point — the maximum — lie, and how can you organize your orchard to blanket the ground in the highest possible number of greenish-yellow Crispins, deep-burgundy Winesaps, tiger-striped Fujis or mottled Honeycrisps?
Be warned. What you’ve just read is a calculus problem.
Students in Sherman Taishoff and Maria Oesterreich’s Calculus AB classes recently learned to resolve questions like this one. It was part of their investigation into a “closed interval,” in which a clear minimum (zero apples) and maximum (the ideal balance between the new-tree factor and the nutrient-stripping factor) are at play.
Mr. Taishoff and Ms. Oesterreich taught this concept using concrete, real-world examples such as the orchard, as well as a hands-on activity. The activity required the girls to create boxes out of single sheets of uniformly sized graph paper, cutting squares of various sizes from the corners and folding the remaining surface into a three-dimensional open-topped box. The girls then developed an equation, using the derivative function in calculus, to determine what the maximum possible volume of such a box could be.
This skill involves complex equations and a grasp of theoretical concepts, but there are plenty of real-world reasons to learn it. Such questions have many economic applications, from maximizing yields to manipulating a market for a product to maximize profit. And if any of these Upper School students should ever live next to someone who gives her an orchard, she’ll know exactly what to do.
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