So earlier, I set up the problem:
Just so you can see how the mind of one mathematician works, here are my notes for when I first came up with the problem:
…which are, of course incomprehensible. Now you know.
Of course, if I were going to put this together for a class, no one would ever see the notes. Instead, they’d see the finished product. It’s the joy of working in the blog format that I can put my notes up just to show the world how badly organized my mind is, and still be able to present a cleaned-up version at the end. I’ll claim it’s inspirational: if he can do math with such a cluttered mindspace, then everyone can do math. (Hmmm…that would make a great title for blog…)
You might remember from geometry (or life) that volume is a three dimensional quantity: you need length, width, and height to describe it. But these drawings are in two dimensions, so you might wonder how we’re going to figure out how far up the riverbed the water flows. The answer rests in something called Cavalieri’s Principle. You can read about it in the Wikipedia article, but the gist is this: The volume of a figure can be found from its cross sections. A good way to visualize this, which the Wikipedia article uses, is to imagine a stacks of coins: the volume of the stack is the same, regardless of whether the coins form a neat stack or whether they are staggered.
Because we’ve assumed uniformity of the cross sections, then the volume of water in the river is going to be proportional to the area of the cross section. Put another way, we don’t need the actual volume; all we need to do is to compare the cross sections. Going back to the stack of coins, if you compare a stack of pennies and a stack of quarters, then provided they have the same height, the volumes of a stack will be proportional to the area of the base coin.
So: In the first house, the cross section of the river is a rectangle, so if the river rises to height above its normal level, then the volume of water is proportional to .
In the second house, the cross section of the river is a rectangle with some triangles attached to it. If the grade is , then filling this up so that the waterline is away from the normal river bank will will require a volume proportional to . I hasten to add that is the horizontal distance; the actual distance will be slightly greater, as it’s measured along the slope.
Because we’ve assumed the volumes of water poured into the rivers are equal, this means we have a relationship between x$, the height of water in the river around the first house, and , the distance from the normal riverbank in the river around the second house:
(If you look at my notes, you’ll see that I have this same formula on the right hand page…except all the variables are different! Lesson Number One: We change variables a lot, or at least I do)
Let’s play around with this equation a little. First, note that we need the square of , which means that if we want to solve this for , we’ll need to do some heavy lifting using the quadratic formula. We’ll do that later, but for now, let’s solve this for instead. First, note that both terms on the left include a factor of , so we can factor it out:
Next, we can divide by to get the formula
Now and are constants: they correspond to the slope of the riverbank up to House 1, and the normal width of the river. Let’s throw in some numbers, just for show.
Suppose feet, (a 1% grade, in road terms). Then
Suppose you’re living feet from the river bank. With a 1% grade, that puts you a mere 1 foot above the normal river level. We let , and we find:
What if your house is 200 feet away from the river? Notice that depends on the square of . This means that increasing will have a disproportionate effect on . In fact, if the water got to you, at a distance of 200 feet from the river, the water would be
above the normal river level…even though you yourself are only 2 feet above it! And if you were 500 feet from the river, in order to reach you, the river would have to be
above the normal river level! When building near a river, distance is more important than height.
We can look at it the other way. If you build a levee, you’re essentially building a wall around a river: in effect, you’re setting up the situation around House 2. Now most levees separate the river from ground that has the same elevation as the river. Here’s my expert drawing of the situation:
A 30-foot high levee is just as effective as putting a minimum of 500 feet between you and the river: any storm that dumps enough water to reach you will overflow the levee.