The Calculus Diaries - Jennifer Ouellette [87]
This saves Dido the trouble of having to calculate the areas of all possible rectangular shapes. She simply graphs the function and then looks to see which values give a slope (derivative) of 0. Even if there are four such spots, instead of one in this particular case, that narrows the possibilities considerably. She can certainly calculate the areas of four possible shapes to determine the best possible width for her planned city.
But remember that the optimal shape is not a rectangle, but a semicircle. To find the true optimal shape, Dido must use the calculus of variations. Just as with the brachistochrone problem, it is necessary to integrate over all possible curves—not just rectangles—to pluck the correct answer from among the infinite hordes. A semicircle with a length of 600 feet has a radius (distance from the circle’s center to its arc) of 191 feet. Since we know a full circle with this radius would have an area determined by πr2, a semicircle has an area determined by ½ πr2. So a semicircle yields an area of 57,296 square feet.
Fans of Virgil know that things did not end well for Dido, queen of Carthage. She rejected the African king’s offer of marriage, only to foolishly fall in love with the wily Aeneas, who wound up in Carthage with his fellow surviving Trojans after the fall of Troy. But Aeneas abandons her to fulfill his manifest destiny of founding the Roman Empire. A heartbroken Dido builds a funeral pyre, curses Aeneas, and falls on a sword given to her by her fickle lover. Aeneas and his men watch the glow of her burning pyre from their departing ship, unaware of Dido’s suicide.
Later in Virgil’s magnum opus, Aeneas travels to the underworld and runs into his former lover’s shade, but she refuses to acknowledge him, still bitter at his abandonment. The poet T. S. Eliot once called this “the most telling snub” in Western literature. I think it shows most clearly that hell literally hath no fury like a woman scorned—particularly a formidable woman like Dido, capable of outwitting an African king with an early conceptual harbinger of calculus, centuries before Newton and Leibniz invented it.
9
Surfin’ Safari
I tried surf-bathing once, but made a failure of it. I got the board placed right, and at the right moment, too; but missed the connection myself. The board struck the shore in three-quarters of a second, without any cargo, and I struck the bottom about the same time, with a couple of barrels of water in me. None but natives ever master the art of surf-bathing thoroughly.
—MARK TWAIN,
Roughing It
Few people are aware that novelist Samuel Clemens—better known by his nom de plume, Mark Twain—was a great admirer of the sport of surfing, or, as he called it in his travelogue Roughing It, surf-bathing. He even tried his own hand at surfing, with predictably dire results: He wiped out and swallowed a hefty helping of salt water for his trouble. I can empathize, as I come up sputtering from a spill for the umpteenth time during my own maiden stab at surfing in Kona, Hawaii. As I scramble back onto my beginners longboard, my self-appointed “surfing coach,” Milton Garces, casually rides a swell over and calls out a snippet of helpful advice: “You might want to move back a bit on your board; you were too far forward that time to ride out the wave!”
He should know; waves of all kinds are his stock in trade, particularly sound waves and water waves. Garces is an acoustic oceanographer at the University of Hawaii-Manoa, specializing in the study of infrasound, aural signals at frequencies that lie below the range of human hearing (20 Hz to 22 kHz). Nature has an entire palette of sounds that play constantly just beyond our ken. Human hearing is rather limited in range, but sound waves exist far beyond it. We can’t hear the ultrasonic shrieks of bats or the ultra-low-frequency waves of acoustic energy (infrasound) employed by elephants or tigers. Wind, water, earthquakes, avalanches, tornadoes, and hurricanes all produce infrasound, as well as