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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.

 

 

Interactive Math: How Trigonometry Can Help You Get Better Sleep

Math is everywhere, even in your dreams! Okay, maybe not literally, but there is some truth to the statement.

During sleep our bodies cycle between different sleep stages, with one full cycle lasting on average 90 minutes. This periodic phenomena can be described by trigonometric functions. In my newest addition to the Interactive Math page I discuss the cosine function that describes a typical sleep cycle, and how you can personalize it to help you get better sleep. Check it out: Sleep Cycle Cosine Model.

 

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.

If Godzilla Were Real, How Much Would He Eat?

(This post originally appeared in Huffington Post’s Entertainment section.)

The newest Godzilla is quite the beast. According to the official source, Legendary pictures, Godzilla is 355 feet tall. That’s about the height of a 40-story building! It’s also the tallest Godzilla yet (according to Wikizilla; yes, there are such sites). So, how much does such an animal need to eat to survive? Let’s do the math to find out.

To answer the question we’ll use Kleiber’s Law: an animal’s basal calorie requirement E (the least amount of calories needed to be eaten in a day to sustain life for a minimally active organism) is related to its mass M (in kilograms) by the equation

E=70M^{3/4}.

We need Godzilla’s mass. There’s no scale I know of that we could use, and I wouldn’t be comfortable asking him directly. Maybe that’s why Popular Mechanics took the easy way out and asked scientists to estimate some of Godzilla’s features. The consensus on his mass: about 160,000 tons, or about as heavy as a cruise ship!

Okay, back to the calculation. Godzilla’s mass in kilograms is 160,000,000 kg, making his daily calorie requirement roughly…

1 million calories!

Maybe this explains why he’s always eating everything in sight!

2014-05-19-gradiotowers-thumb

(And also why Hollywood — and the original creator of Godzilla — made him eat nuclear energy for food.)

Truth be told, I’m not sure how applicable Kleiber’s Law is to Godzilla. It does, however, accurately describe the relationship of mass to energy requirements for everything from cells to elephants. That’s a pretty amazing feat for just one equation. And, aside from the absurdly large appetite Godzilla has, it’s the one other thing I hope you’ll take away from this article.