The Calculus Diaries - Jennifer Ouellette [86]
Ah, but just how do we know that Dido’s semicircle did indeed enclose the largest area given the length of that long thin strip? Calculus, of course—specifically, we must solve a maximization problem using the calculus of variations. Let’s start with a simpler idealization to demonstrate the basic method. We’ll assume that Dido’s strip of oxhide is 600 feet long, and she wants to enclose the largest possible rectangular area in which the seashore provides a boundary along one side. Even in this simplified version, there are many different possible permutations she could make with that 600 feet of oxhide: long and narrow, tall and thin, and everything in between. What shape is the likely candidate for giving her the optimal square footage?
We have to start somewhere, so let’s take the shape of a square as a point of reference. By definition, this means that Dido would need 200 feet of oxhide for each of the three sides, with the shoreline of the Mediterranean Sea serving as the fourth side. That gives us an area of 40,000 square feet, as area depends on length and width. However, we have no way of knowing (yet) whether this is indeed the optimal shape. So we begin varying the shape ever so slightly in different directions. For instance, if Dido arranges her oxhide to measure 201 feet down the width of two opposite sides and 198 feet across the length, she would have an area of 39,798 square feet—slightly less than a perfect square. Dido decides to test this further, and adjusts her dimensions in a different way. This time, the two opposite sides measure 199 feet in width, and the third side measures 202 feet across. The answer: 40,198 square feet. Clearly the square will not give her the most possible area.
The crucial point is that the question posed has to do with change in area, not simply the static values of the area—that way lies madness, for we would be randomly computing areas for different configurations in hopes of stumbling on exactly the right one. It is far more useful to consider all possible areas created by all possible configurations (i.e., an infinite number). We have now seen countless times how the derivative applies to any case where a change in one quantity produces a corresponding change in another quantity; the derivative measures that rate of change.
Dido can reduce the problem to a simple function: Knowing she has 600 feet of oxhide, once she chooses a width for her enclosure (w), the remaining oxhide will be evenly divided to make up the length of the plot of land. How do we translate this into an equation? We know that area equals width multiplied by length. So given the variables for width (w) and length (L) of the configuration, as well as the total amount of oxhide (600), we come up with the function 600w − 2w 2. This is the function she would use to determine what the area would be for any given configuration she chose. Graph that function by plugging in various values for w—ranging from a width of 0, to a width of 300—and you get a pretty curve (the “face” of Dido’s function). This greatly narrows the possible solutions.
Now all we need to do is find a spot along that graph where the rate of change is 0. Recall that the slope of the tangent line along a curve is equivalent to the derivative. The place where the derivative is 0 will therefore be at the very top of the curve, where the tangent line is a horizontal line and hence has no slope. And that point occurs where w = 150 feet. Ergo, Dido’s plot of land should be 150 feet wide to get the maximum possible rectangular area (45,000 square