Dear Mr. Avering, Get a grip. Eve is not worth mooning over; she’s an evil, manipulative itch that you can’t scratch. She destroyed the old dipstick, and she lied about the shape of your tank. And I’m going to prove it. I’m going to show you that your tank is not hemispherical. And I’m going to show you that building a new dipstick -­‐-­‐ for your parabolic tank -­‐-­‐ is as easy as pie. Or rather, it’s as easy as 50-­‐pi-­‐H2; that’s the volume (in cubic feet) of material in your tank when it’s H feet deep. I’ve drawn a picture below. It shows a superimposed pair of cross-­‐sections of two tanks: a hemispherical tank and a parabolic tank. You can see that the main difference is that the hemisphere is always wider, except at the very top and the very bottom. Because that’s the only difference, it might be helpful to set up two formulas to describe the widths. It’s easiest to do the math if I let w stand for the width from the center (that is, measured out from the dipstick to the wall of the tank). I’ll measure both w (width) and h (height of the material) in feet. Figure 1: In this figure, we see that a parabolic tank is thinner than a hemispherical tank. For a hemisphere, width and height are related by that old circle formula, w2sph + (h-­‐
100)2 = 1002. For a parabolic tank, width and height are related by a different formula: 100*h = w2par . The first looks like the “(x-­‐x0)2 + (y-­‐y0)2 = r2” circle you learned in geometry; the second one is sort of like the “y=x2” that you might remember as parabolas from your pre-­‐
calculus. The “100”s in each of these formulas were included to stretch the or move the graphs so that they match your tank at the bottom and at the top. At the bottom, both width and height are 0; at the top, both width and height are 100 feet. The following table shows you that both these formulas line up with the measurements at the top and bottom. hemisphere
formula relating w & h w2sph + (h-­‐100)2 = 1002 top (w=100, h=100) 1002sph + (100-­‐100)2 = 1002 bottom (w = 0, h = 0)
02sph + (0-­‐100)2 = 1002
Table: Both the hemisphere and the parabola pass through the correct points at the bottom and top of the tank: (0,0) and (100,100). That’s why Eve was able to fool you for a little while: on the parts you could measure with a long ruler, both of these shapes could be right. But you helped a lot by measuring both the volume and the depth of how much material you put in the tank so far, and that’s how we’ll show that Eve was doing you wrong! We’re going to compute a formula for volume by adding up thin layers of material, one layer on top of another, like your grandpa’s favorite 16-­‐layer cake. To do this, I have to assume that each new, thin layer is a perfect circle -­‐-­‐ meaning that your tank isn’t lopsided or uneven. I’m also going to assume that each new, thin layer is perfectly level. If you’re storing boulders or old office chairs that pile up in uneven heaps, then all these computations are garbage! But if you’re storing something liquid, read on! Since each layer is going to be like a perfect, flat, circle, we can use the formula for the area of a circle to approximate the volume of a thin layer: Volumelayer ~ π (radius)2 * thickness of the layer. The “radius” here is just the width of the tank, measured from the center dipstick. That is, it’s just the width (w) we used above. So we’ll substitute in the relationships between width and height that we wrote on the previous page. For a hemisphere, the volume of a layer is Volumesph-­‐layer ~ π (wsph)2 * thickness of the layer. = π [1002-­‐(h-­‐100)2] * thickness of the layer. For a parabolic tank, the volume of a layer is Volumepar-­‐layer ~ π (wpar)2 * thickness of the layer. = π [100h] * thickness of the layer. If we use a thickness of 1 foot, we can use the formulas above to get a rough estimate for how much is in the tank up to 10 feet high, one layer at a time. In the table below, I calculated the approximate volume for the bottom 10 layers, using both formulas. I used the middle height in each layer, so I could get a closer approximation. height of the 1-­‐ft layer
volume of the layer
vol
h=?
in a hemisphere
in a
h=?
π*[1002-­‐(h-­‐100)2]*1
π*1
0.5
313
157
1.5
935
471
2.5
1551
785
3.5
2161
110
4.5
2764
141
5.5
3361
172
6.5
3951
204
7.5
4536
235
8.5
5114
267
9.5
5685
298
subtotal
30371
157
Table: Using flat disks with a thickness of 1 foot to estimate the volume at the bottom of a hemisphere and a parabola. Of course, this is just an estimate. To get a better answer, we’d take incredibly thin layers (but also an incredibly large number of those layers) and add them up. In calculus, doing that is called “integration”, and the symbol for that looks like the “S” of a sum. There is also a technique to find a simple formula that adds these many skinny layers up. I’ll show you the notation that mathematicians use, just in case you want to impress your friends, but all you really need to know is the simple formula that comes from that fancy notation. For a hemisphere, the volume (in cubic feet) for material up to H feet high is shown symbolically as Volumesph(H) = §0H π [1002-­‐(h-­‐100)2] dh, and the simplification symbolic stuff is just Volumesph(H) = 314*H2-­‐1.047 H3. So more precisely, you’ll see that the bottom 10 feet of a hemisphere have about Volumesph(10) = 31,400-­‐1,047 =30,353 cubic feet of material. That’s not far off from the estimate we had above, but it’s very far from what you measured yourself! That’s how I know Eve was lying to you! So let’s do the same thing for the real tank: the parabola. The symbol for summing up the volume of all those thin layers up to a height of H feet is Volumepar(H) = §0H π [100h] dh, And the simplification that you can actually use to compute volumes is Volumepar(H) = 50*π*H2, or even more usefully, Volumepar(H) = 157.08*H2. And we can use this to give us the markings on your dipstick for any height H. Here is a table with markings every 10 feet. Height
10
20
30
40
50
60
70
80
Volume
15,708
62,832
141,372 251,327
392.699
565,487 769,690 1,005
Table: The markings of volume for the dipstick, if you mark it every 10 feet, using the formula Volumepar(H) = 157.08*H2. That’s all you need to figure out the volume of material, or how to mark your dipstick. Get back to working with your granddad, and leave that witchy woman alone! Sincerely, A concerned calculus student