The Calculus Diaries - Jennifer Ouellette [46]
If we know the combined weight of the passengers and the log we are riding in, the volume of our log, and the collective density (in units of grams per cubic centimeters), we can divide the total weight by the total density to get our volume in cubic meters. We also need to know the density of water; a quick Google search reveals that one liter of water has a density of 1 kilogram. Now we multiply the volume of our log and its passengers by the density of the water to find the volume of water displaced. Those hollow plastic logs hold six riders of varying weights. Assuming an average weight of 150 pounds per passenger (150 × 6, plus the weight of the log itself), that gives us a pretty substantial volume—and a substantial displacement of water when we hit the bottom of that final plunge. No wonder we’re completely soaked by the ride’s end.
Sodden jeans and sneakers are not a pleasant sensation. It is a cool, cloudy day for Anaheim and late enough in the afternoon that our clothing takes longer to dry than it would on a warmer day. While we wait, Sean explains that there is a calculus problem in our current plight: The rate at which our clothing dries—that is, the rate of evaporation of water from the fabric—forms an exponential decay curve. It is similar to the rate at which a cup of hot coffee cools until it reaches thermal equilibrium with its surroundings.
The coffee cools off very quickly at first, but as it gets closer to thermal equilibrium, that rate of cooling slows down and eventually levels off. This is because the amount of heat lost is proportional to the temperature of the coffee: It is determined by the ratio of the excess heat to the lower temperature limit—how cool the coffee can get, usually ambient room temperature. So as the coffee cools down and gets close to room temperature, there is less excess heat and thus a smaller ratio between the two variables. And the rate of cooling levels off.
The same thing happens with the evaporation of the moisture in our clothing. Plot the rate of evaporation as a function of time, and you can see this in the resulting curve: There is a steep drop initially, followed by a gradual leveling off. The alert reader will note that because we are dealing with a rate of change, we must be taking a derivative. That means we can find an answer to the question, “How fast is the water in our clothes evaporating at x time?”—a form of the velocity function, similar to determining our instantaneous speed in chapter 2—by finding the slope of the tangent line along that particular point on our curve.
We experience this exponential decay curve firsthand and soon find ourselves wondering, Will we forever be slightly damp? It is beginning to feel that way, and we have dinner reservations in an hour at the Blue Bayou restaurant—the sole fine dining establishment in Disneyland, situated just inside the Pirates of the Caribbean ride. We end up squishing our way over to the gift shops in New Orleans Square in search of a change of clothes, where the sales clerk assures us this happens all the time. They do a brisk business, thanks to Splash Mountain.
And thus we find ourselves, an hour or so later, seated in the Blue Bayou’s fake outdoor grotto in matching Pirates of the Caribbean hooded sweatshirts, my outfit completed by a jaunty newsboy cap with a skull-and-crossbones motif to hide my hopelessly tangled hair. By this time Sean is very much in need of a drink, and the Tinkerbell Fruit Punch with a fairy light garnish simply isn’t going to cut it. Alas, there is no alcohol to be found in the Magic Kingdom, depriving us of a prime opportunity to work out the calculus of inebriation. (Oh yes, it can be done.) We content ourselves with a sugar rush instead and split the signature dessert: a boat-shaped “cookie” with an edible sail featuring the obligatory skull and crossbones. It is a pirate’s life, indeed.
5
Show Me the Money
It is clear that economics, if it is to be a science at all, must be a mathematical