For the sake of argument, say that you happen to be standing on the edge of a beach off the coast of Ireland, directly facing a little island called Craggy Island. Fans of the British television series “Father Ted” will be getting excited at this point, since Father Ted and his fellow renegade priests were exiled to Craggy Island due to some unspecified but nevertheless unsavoury behaviour. But more on that some other time.
Where was I? Oh yes! For some inexplicable reason, you have been seized by a powerful urge to kayak to the island, but you don’t know how far away it is. Assume you’ve never seen “Jaws”. How could you determine the length of your impending voyage?
Personally speaking, I would just call the Craggy Island Department of Tourism and Limpets (1-877-LIMPETS), and ask them how far it is to their island. But maybe there’s no answer when you call and maybe you just don’t like taking the easy way out. So now what?
Well…if you happen to know how fast the lighthouse beacon is rotating, and if you happen to know how fast the beam is sweeping toward you when it hits the (eerily-straight) shore 10 kM north of where you’re standing, you could say to yourself: “This sounds like a related-rate problem! I might be able to use Calculus to solve it!”
Since you have nothing better to do, you resist the urge to start paddling, so you fly home and spend the next few nights covering page after page with chicken scratch, basically re-deriving Calculus from first principles, since you haven’t taken it for 38 years or so. (The picture which should pop into your head at this point is that of a large beetle flipped over onto its carapace in front of a blackboard, feebly waving a piece of chalk clutched in its foreclaw.)
Finally, your son (who oddly enough happens to be taking Calculus at school) takes pity on you after witnessing your struggle and says, “Dad, why don’t you just Google it?”
Since you were born well before Al Gore invented the Internet, you look at him with a dumbfounded expression and reply, “What the heck would I Google?”
He regards you with a sorrowful expression and says, “I dunno. Google is pretty clever. Try typing Calculus lighthouse problem.”
You dutifully follow this directive and to your undying amazement, this search phrase returns a long string of hits, and one of them even refers to Craggy Island!
From there it’s just a short hop to a YouTube video clip (https://www.youtube.com/watch?v=0QyTGpeZKrg) made by an endearing fellow named Bart Snapp who solves your exact problem right before your very eyes! You really should watch this clip, mostly because I took the trouble to transcribe the intro almost word-for-word, but also because you will find yourself swept away by Bart’s patently obvious love of teaching in general, and Calculus in particular, and also because this guy is great at reading out loud. I quote:
“Hello there! Now we’re going to do a problem (waves hands energetically) about a beacon in the ocean, also known as a lighthouse of sorts. But we’re going to call it a beacon. All right?
“All right! Let’s see the problem! (Reads problem enthusiastically and eloquently.)
“All right. So we have our problem and now we have to… (he pauses for dramatic effect)…Draw a picture! (Bart starts sketching rapidly on a whiteboard)
“All right. So we have a, we have the shore here, and the shore’s supposed to be straight. (Draws more-or-less straight line)
“Well. Well that’s straight enough I guess. Here’s point A. (Draws the beacon on a line perpendicular to point A, and presses on)…the beacon has some light that is shining and let’s see…(adds some more notations like dϴ/dt and dx/dt to his diagram).
“And the water…This is all supposed to be water here. (Draws blue squiggles.) That’s great.”
And really, it is great because at that point you see that the beacon, Point A and the point where the beam hits the shore 10 kM north of you form a triangle, and you can relate the rotational speed of the beacon (in radians/sec of course) to the sweep rate of the beam along the shore through trigonometry!
From there it’s only a matter of a couple more days of calculations until you figure out the lighthouse is 10 kM straight out from where you’re standing. You can easily handle a 10 kM paddle but then you find out that the last person who tried it was eaten by sharks. What should you do?
The first thing you should do is ask yourself whether this whole eaten-by-ravenous-sharks while-attempting-to-paddle-to-Craggy Island thing is true or is it an Urban Legend? And does Craggy Island even exist, or is it the product of the imagination of a couple of half-baked Irish writers named Arthur Matthews and Graham Linehan?
This is important because Urban Legends are everywhere these days, thanks to Al Gore, and you just can’t be too careful. Consider the story I read recently about an intoxicated Marine in the state of Kansas who was arrested after a failed attempt to foil his car ignition interlock by having a raccoon breathe into it.
Right away you have to be suspicious that this is an Urban Legend because there are no raccoons (or Marines for that matter) in the state of Kansas. Actually, I’m lying. I made up the Kansas part, and if you consult Google, you will discover that Kansas is literally teeming with raccoons. I quote from the Kansas Department of Wildlife, Parks and Tourism website: “Highly intelligent and adaptable, the raccoon (Procyon lotor) is one of our most abundant furbearers. “ But all that aside, your first clue that the story is B.S. is that everyone knows Marines don’t drink.
Anyway, the point is that you have to be careful not to waste your time on drivel like the story of The Raccoon and The Breathalyzer. Instead I want you to recall this poem by British artist, illustrator, musician, cookbook author and all-around oddball Edward Lear (1812-1888).
With apology to all the people of Britain, except Prince Charles:
The Owl and the Pussycat did some maths, with the aid of Barton Snapp
They related dtheta to dx (by dt) but they found themselves in a trap
They set forth for the Isle but in a short while, the water began to boil
Attacked by a shark, in the cold and the dark, off shuffled their mortal coil(s).
Well, it’s time to shut this thing down. I think I’ll heat up a plate of leftover limpets and then get ready for my paddle to Craggy Island first thing in the morning. But just one nagging thought remains: I may need a bigger boat.