This is a solution of sorts, but we can simplify it even further. First, there’s a basic rule when dealing with exponentials that when we have xa+b, we can rewrite it as xa⋅ xb. So we can rephrase the equation yet again as y= ekx⋅ec.
Finally, we can rewrite eC as simply C since both are constants; that lets us combine them into one big constant. After all is said and done, we end up with the function y = Cekx. Whew! This is the equation that encodes the answer to the question we’ve posed.
If you’re anything like me, by now your head hurts, and you just want to be done with it. But remember the zombies! We need to figure out how fast the zombie infection is spreading, because our very lives may depend on it. As we learned in chapter 2, the solutions to differential equations aren’t specific numbers; they are new equations. And this particular equation holds the key to determining the growth rate of the zombie population.
What does our new equation tell us? Well, C stands for the initial population of zombies (a constant number that doesn’t change), while y now stands for the total zombie population after a certain amount of time (denoted by x) has passed. We’ve got Euler’s constant (e) lurking around as well, but we need not worry about it just yet. That leaves k, which will tell us the rate of zombie infection.
We need to figure out the value of k. We start with the initial zombie population: Let’s say on day 1 there were 19 people who ate contaminated hamburger and turned into zombies ravenous for tasty brains. Ten days later, we count again and discover their ranks have swelled to 193 zombies. That’s all the data we need to solve for k, using our handy little formula: y = 193 (new number of zombies), C = 19 (beginning number of zombies), and x = 10 (days that have passed): 193 = 19e10k.
Next we take a series of steps to simplify and solve this equation. Since k is the value we want to find, we must isolate k on the right side of the equation. First we divide by 19 on both sides of the equation: .
Now we need to get rid of that annoying exponential function, which we do by reintroducing the natural logarithm: .
Next we divide both sides of the equation by 10 to isolate .
Finally, we get our answer: k = 0.2318. Aren’t you glad you invested in that calculator? Brace yourself, because we’re not quite done yet.
Now we can plug the values for C and for k into our handy little equation: y = 9e 0.2318x. With this information, we can determine the number of zombies there will be any number of days in the future, just by varying the x factor. Voila! We have a truly predictive model; that is the beauty of the mathematical function.
For instance, how many zombies will there be after thirty days? Make x = 30, and we get 19,914 zombies. Hmmm. That’s some serious exponential growth. I’m sure our merry band of zombie killers would agree: We will be outnumbered very quickly. Evacuation is definitely in order.
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GENERAL ONLINE RESOURCES
Sites:
Albion Research
American Dietetic
Association
Anecdotage.com
Arcytech.com
ArXiv
BBC News Online
The Calculus Page
Cracked.com
CrapsCenter.com
eHow.com
Health Map
How Stuff Works
Investopedia
Los Angeles Wheelmen
MacTutor History of
Mathematics Archives,
University of St.
Andrews (Scotland)
Mouse Planet
National Curve Bank
Reducingstereotypethreat.org
The Straight Dope
Surfing Legends
Surfing for Life
U.S. Centers for Disease
Control
Walkability Score
Wikipedia
William Poundstone.net
Wired.com
Wolfram MathWorld
World Health Organization
Blogs:
3 Quarks Daily
Aetiology
BldgBlog
Built on Facts
Calculated Risk
Cocktail Party Physics
Confounded by Confounding
Cosmic Variance
Damn Interesting
Doctor Housing Bubble
Effect Measure
The Frontal Cortex
Good Math, Bad Math
Gravity and Levity
In the Dark
Io9
Laelaps
Mike the Mad Biologist
Neurophilosophy
Not Exactly Rocket Science
Physics arXiv Blog
The Quantum Pontiff
Retrospectacle
Sciencegeekgirl
Southern Fried Science
Starts with a Bang
Swans on Tea
Tiny Cat Pants
Uncertain Principles
INDEX
acceleration
in Devil Dive ride
in Space Mountain ride
in surfing
in Tower of Terror ride
&n
bsp; Addison, Joseph
Aeneid (Virgil)
Agboola, Bisi
Agnesi, Maria Gaetana
algebra
geometry linked to
Amber Spyglass, The (Pullman)
American Association for the Advancement of Science
amplitude
amusement parks
Disneyland, see Disneyland
Six Flags
Analytical Institutions (Agnesi)
Andersson, Lennart
ants
Apollonius of Perga
arches
Archimedes of Syracuse
death ray device of
golden wreath and
The Method
architecture
arches in
Gaudi and
Ochsendorf and
Aristotle
Attenborough, David
Atwater, Wilbur
Auerbach, Erich
Bacon, Dave
Bacon, Kevin
Bacon, Roger
Bandel, Hannskarl
Banting, William
Bardi, Jason
baseball
Batlló, Josep
BBC News
Begley, Ed, Jr.
bell curve (Gaussian distribution)
Bergamini, David
Berkeley, George
Berlinski, David
Bernoulli, Daniel
Bernoulli, Jakob
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 25