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Everyday Calculus Reader Question

Reader S. Cho wrote in with the following question:

Hello! I have read your book, Everyday Calculus. It is very interesting and gives me a knowledge for math. I realized that math is all around us. But I have two questions in the book. First, I met the formula “r(V)=k sqrt(P_0 l) over V” in chapter 1. However I couldn’t find this formula in any other books. Could you give me the source of it or a reference book? Second, does P_0 mean the electric power? I’m waiting for your reply. Sincerely yours.

 

The formula r(V)=k\frac{\sqrt{P_0l}}{V} comes from the combination of the three formulas:

P_0=IV,         R=\rho \frac{l}{\pi r^2},          V=IR.

The first formula relates the voltage (V) and current (I) produced by a power source (P_0). The second is Pouillet’s Law, which relates a power line’s resistance to the flow of electric current R to the power line’s length l and radius r (I’ve assumed the power line is a long cylinder). The constant \rho is called the electrical resistivity. The last formula is Ohm’s Law.

Now, if we solve Ohm’s Law for I and plug that into the first equation we get

P_0=\frac{V^2}{R}.

Then, using Pouillet’s Law we get

P_0=\frac{V^2(\pi r^2)}{\rho l}.

Solving this for r^2 yields

r^2=\frac{\rho}{\pi}\frac{P_0l}{V^2}.

Taking the square root of both sides and letting k=\sqrt{\rho/\pi} gives

r=k\frac{\sqrt{P_0l}}{V}.

In the first chapter of Everyday Calculus I explain how this equation spelled doom for Thomas Edison. His early power grid operated at a fixed voltage of 110 Volts, which meant that as the length of the power line (l) increased, the power line’s radius (r) got bigger too. To avoid large power lines hanging over pedestrians (clearly a danger for anyone walking underneath them) Edison therefore had to build his power plants very close to his customers. This represented a huge cost, and limited the early usage of electricity. But the same equation also tells us that we can avoid this issue by using much larger voltages, which is what’s done today. These high voltages coming out of the power plants–in some cases nearly 1 million Volts–are reduced to the familiar 120 Volts we see in homes using transformers, which are the result of other neat physical principles (and math) at work.

 

 

More Snow for Boston, Says Calculus

In a recent Huffington Post article I calculated the probability that Boston will get even more snow than its current historical record. The function I ended up with was

P(s)=e^{-0.079s}, \

where is the additional inches of snow beyond the 45.5 inch record set on Saturday (February 15th). Let me explain how I got that function.

First, Some Background in Probability

Let’s start with some basics. The total snowfall in any given month is a variable. But it’s not your “usual” kind of variable, it’s what we mathematicians call a “continuous random variable.” These type of variables have a function associated with them—called a “probability density function” or “PDF”—that tells us the probability that the random variable’s value will fall within a certain range (the “continuous” part means that the values of the random variable can be any real number). In our case the random variable of interest is the total snowfall in February in Boston, which I’ll denote by S, and it’s associated PDF, which I’ll denote by p(x). Then the probability that S is between, say, 45.5 inches and 45.5+h, where h is small (like, way less than 1), is approximately

P(S=45.5) \approx p(45.5)h. \

To approximate the probability that S is between, say, 45.5 and 46.5 we’d need to add up many terms like the one on the right-hand side. One such approximation is

P(45.5 < S \leq 46.5) \approx p(45.5)(0.1)+p(45.6)(0.1)+\ldots+p(46.4)(0.1). \

(This is the case when all h’s are equal to 0.1). If we want the exact answer we’d need to add up infinitely many terms, each corresponding to an h-value that is infinitesimally small. The way we do that in calculus is by integrating. So, in calculus speak,

P(45.5<S \leq 46.5)=\displaystyle\int_{45.5}^{46.5}p(x)\,dx. \

Now Back to Boston’s Snowy Month

If you’ve made it this far one thing is clear: we can’t calculate anything without the PDF. That’s where the data linked above comes in. By downloading the total snowfall column in the data into a spreadsheet we can create the histogram below.

PNG2

 

This histogram tells us how frequently (in the 95 years between 1920 and 2014) the Feburary snowfall total was between 0 and 5 inches, 5 and 10 inches, etc. (For example, the first bar says that the total snowfall was between 0 and 5 inches 26% of that time.) The black curve is the exponential function

f(x)=0.511e^{-0.079x}, \

where is the total snowfall (in inches). This curve is Excel’s best fit to the data. It’s not perfect, but it does a better job than other fits (like a linear function).

To make f(x) into a PDF we need to make sure that the probability that S is between zero and infinity is 1. (Roughly speaking, this expresses the fact that all probabilities must add to 1.) In calculus jargon, this means that we first need to calculate the integral of f(x) between zero and infinity and then divide f(x) by that value. And since

\displaystyle\int_{0}^{\infty}0.511e^{-0.079x}\,dx=\frac{0.511}{0.079} \approx 6.416. \

Our PDF—which I’ll call p(x)—is therefore

p(x)=\frac{f(x)}{6.416}=0.079e^{-0.079x}. \

(This is the PDF of an exponential distribution, a well known PDF from probability theory that has many applications to business, physics, and engineering—see here for more.)

We’ve found our PDF…woohoo! We can now calculate the probability that S—the total snowfall amount in February in Boston—will be between zero and some other number y. As before, that’s just the following integral:

P(0 \leq S \leq y) = \displaystyle\int_{0}^{y}0.079e^{-0.079x}\,dx. \

This is a pretty straightforward integral to calculate using a technique called u-substitution. The answer is

 1-e^{-0.079y}. \

We’re almost done, I promise. The last step is to use the fact that 45.5 inches have already fallen. So, given that 45.5 inches of snow have already accumulated, what is the probability that s more inches will fall? Well, if we denote by y the total snowfall (i.e., y=s+45.5), then in math-speak we want to calculate

P(S>y|S>45.5). \

This is an example of a “conditional probability,” and by the laws of probability, this simplifies to

1-P(S \leq y|S>45.5)=1-\frac{P(45.5<S \leq y)}{P(S>45.5)}, \

and in terms of integrals becomes

1-\frac{\displaystyle\int_{45.5}^{y}0.079e^{-0.079x}\,dx}{e^{-0.079(45.5)}}. \

Finally, calculating and simplifying gives

1-\frac{e^{-0.079(45.5)}-e^{-0.079y}}{e^{-0.079(45.5)}}=e^{-0.079(y-45.5)}=e^{-0.079s}. \

This is the P(s) formula I gave at the start of the article. It’s amazing what the internet, math (oh, and Excel) can accomplish. Pretty neat huh?

Interactive Math: Calculus of Coffee Cooling and Secant Lines to Tangent Lines

New Additions to the Interactive Math Page

I’ve added two new links to the Interactive Math page:

 

  • Secant lines to tangent lines. Learn how secant lines (lines through two points on a graph) are used to introduce the derivative in calculus. You’ll get to interact with a graph and see how a tangent line (whose slope is the derivative) emerges by making the two points a secant line passes through closer together.

 

 

Both additions are related to content from my book, Everyday Calculus. Visit the Interactive Math page to find out more.

These Two Numbers Make Spring Possible (and the other seasons too)

(This post originally appeared on Princeton Press’ blog.)

March 20th. Don’t recognize that date? You should, it’s the official start of spring! I won’t blame you for not knowing, because after the unusually cold winter we’ve had it’s easy to forget that higher temperatures are coming. But why March 20th, and not the 21st or the 19th? And while we’re at it, why are there even seasons at all? Read on to find out the answers.

The answer has to do with 2 numbers. Don’t worry, they’re simple numbers (not like pi [1]). Stick around and I’ll show you some neat graphs to help you understand where they come from, and hopefully entertain you in the process too.

The first star of this show is the number 92 million. No, it’s not the current Powerball jackpot; it’s also not the number of times a teenager texts per day. To appreciate its significance, have a look at our first chart:

f1
Figure 1: The average surface temperature (on the vertical axis) of the planets in our solar system sorted by their distance from our sun (the horizontal axis). From left to right: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto (technically, Pluto lost its planet status in 2006).

 

That first planet on the left is Mercury. It’s about 36 million miles away from the sun and has an average surface temperature of 333o. (Bring LOTS of sunscreen.) Fourth down the line is the red planet, Mars. At a distance of about 141 million miles from the sun, Mars’ average temperature is -85o. (Bring LOTS of hot chocolate.) We could keep going, but the general trend is clear: planets farther away from the sun have lower average temperatures [2].

If neither 333o nor -85o sound inviting, I’ve got just the place for you: Earth! At a cool 59o this planet is … drumroll please … 92 million miles from the sun.

We actually got lucky here. You see, it turns out that a planet’s temperature T is related to its distance r from the sun by the formula , where k is a number that depends on certain properties of the planet. I’ve graphed this curve in Figure 1. Notice that all the planets (except for the pesky Venus) closely follow the curve. But there’s more here than meets the eye. Specifically, the formula T=k/sqrt{r} predicts that a 1% change in distance will result in a 0.5% change in temperature [3]. For example, were Earth just 3% closer to the sun—about 89 million miles away instead of 92 million—the average temperature would be about 1.5o higher. To put that in perspective, note that at the end of the last ice age average temperatures were only 5o to 9o cooler than today [4].

So our distance from the sun gets us more reasonable temperatures than Mercury and Mars have, but where do the seasons come from? That’s where our second number comes in: 23.4.

Imagine yourself in a park sitting in front of a bonfire. You’re standing close enough to feel the heat but not close enough to feel the burn. Now lean in. Your head is now hotter than your toes; this tilthas produced a temperature difference between your “northern hemisphere” and your “southern hemisphere.” This “tilt effect” is exactly what happens as Earth orbits the sun. More specifically, our planet is tilted about 23.4o from its vertical axis (Figure 2).

f2
Figure 2. Earth is tilted about 23.4o from the plane of orbit with the sun (called the ecliptic plane).

 

Because of its tilt, as the Earth orbits the sun sometimes the Northern Hemisphere tilts toward the sun—roughly March-September—and other times it tilts away from the sun—roughly September-March (Figure 3) [5].

f3
Figure 3. Earth’s tilt points toward the sun between mid-March and mid-September, and points away from it the remaining months of the year. The four marked dates describe how this “tilt effect” changes the number of daylight hours throughout the year. Assuming you live in the Northern Hemisphere, days are longest during the summer solstice (shorter nights) and shortest during the winter solstice (longer nights). During the equinoxes, daytime and nighttime are about the same length.

 

Now that you know how two numbers—92 million and 23.4—explain the seasons, let’s get back to spring in particular. As Figure 3 shows, there are two days each year when Earth’s tilt neither points toward nor away from the sun. Those two days, called the equinoxes, divide the warmer months from the colder ones. And that’s exactly what happened on March 20th: we passed the spring equinox.

Before you go, I have a little confession to make. It’s not entirely true that just two numbers explain the seasons. Distance to the sun and Earth’s tilt are arguably the most important factors, but other factors—like our atmosphere—are also important. But that would’ve made the title a lot longer. And anyway, I would’ve ended up explaining those factors using more numbers. The takeaway: math is powerful, and the more you learn the better you’ll understand just about anything [6].

 


 

[1] The ratio of a circle’s circumference to its diameter, pi is a never-ending, never repeating number. It is approximately 3.14.

[2] Venus is the exception. Its thick atmosphere prevents the planet from cooling.

[3] Here’s the explanation for the mathematically inclined. In calculus, changes in a function are described by the function’s derivative; the derivative of T is T'(r)=-kr^{-3/2}/2. This tells us that for a small change dr in r the temperature change dT is (-kr^{-3/2}/2)dr. Relative changes are ratios of small changes in a quantity to its original value. Thus, the relative change in temperature, dT/T, is

frac{dT}{T}=left(-frac{kr^{-3/2}dr}{2}right)left(frac{r^{1/2}}{k}right),

which is minus 0.5 times the relative change in distance, dr/r. The minus sign says that the temperature decreases as r increases, confirming the results of Figure 1.

[5] Just like in our thought experiment, the Southern Hemisphere’s seasons are swapped with our own; when one is cold the other is warm and vice versa.