Larry Gladney is Associate Professor of Physics and Dennis DeTurck is Professor of Mathematics, both at the University of Pennsylvania.

The most useful functions for science are exponential functions. They have an incredible number of applications -- we present two in this module.

Larry Gladney is Associate Professor of Physics and Dennis DeTurck is Professor of Mathematics, both at the University of Pennsylvania.

The most useful functions for science are exponential functions. They have an incredible number of applications -- we present two in this module.

Read about Thomas Malthus, for whom the Malthusian Model is named.

The simplest model of population growth assumes essentially that the adult (female) members of a population reproduce at a steady rate, usually as fast as they can. This implies that births increase the population at a rate proportional to the population. Similarly, a certain proportion of the population dies off every year -- so deaths decrease the population at a rate proportional to the population. If the proportionality constant for the birth rate is greater than that for the death rate, then the population increases, otherwise it decreases. In this simple situation, the population either increases or decreases exponentially.

In this project, we examine the exponential growth of a population of fast-breeding organisms -- cockroaches. Before moving on to the problems below, view the movie clip that illustrates the situation. [This clip was found to be inoperable on 3/20/2012. The link was removed. Ed.]

Cockroaches are pretty large bugs that breed *very* quickly. For the purposes of this project, we will assume the following:

- Cockroaches breed quickly enough that their population doubles every minute.
- It takes 10 cockroaches to cover one square inch of the ground.

Assume that you start with a population of 1 cockroach (don't ask how it reproduces). Then the cockroach population is 2* ^{t}*, where

- How many cockroaches are there after 10 minutes?
- What total area do they occupy after 10 minutes?
- How many cockroaches are there after 15 minutes?
- What total area (in square
*feet*) do they occupy after 15 minutes? - How long until you have 1,000,000 cockroaches?
- How long until there are enough to cover the floor of an 8' by 12' kitchen?
- How long until they cover the floor of a typical 2000 square foot house?
- How long unti they cover an area the size of the city of Philadelphia (according to the 1993 World Almanac, 136 square miles)?
- How about an area the size of Pennsylvania (44,820 square miles)?
- How about an area the size of the United States (3,539,289 square miles including all 50 states plus DC)?
- How about all of North America (as shown at the end of the video -- 9,400,000 square miles)?
- How about covering the world (57,900,000 square miles)?
- Use a watch to get timings from the video on how long it takes for the population to double (at the beginning of the video) and how long until North America is covered. Is the video realistic in this regard?

Cerenkov light from gamma rays due to Cobalt-60 decay in water |
Pennies -- for simulating radioactive decay |
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In talking about problems like population growth, we needed to learn about the exponential function. This function is useful for describing many very different observations in science. In fact, any change in the number of objects which depends *proportionally* on the number of objects present will be described by an exponential. Clearly, population growth depended on this since the number of people that can be born in a year depends on the number of people able to bear children during that year.

In this experiment, you will consider a similar case, that of *radioactive decay*. As you know, all materials are made of atoms. If the material is radioactive, that means that some of its atoms are continually dying, or, more accurately, they are being transformed into some other type of atom. The probability that any given atom in the material will decay is the same as for all atoms and this probability does not change with time, i.e. the chance that an atom will decay in the next second is unaffected by the fact that it did not decay a second ago. Clearly the number of atoms decaying in one second depends on the number of atoms you start with, but the chance of any individual atom decaying in a given time period is always the same. These are the same conditions as for population growth (which assumes that everyone has the same chance for having babies no matter what particular time period we choose) except that here the population is decreasing rather than increasing with time.

The description of why atoms decay is quite complicated, involving quantum mechanics and nuclear physics. The amazing thing is that we can accurately mimic, in a quantitative way, what happens during radioactive decay with just a few pennies. How? Well, start by noting that when you toss a single coin, the chance that it will land tails up is about 50%, the same as the chance that it will land heads up. It always lands either heads or tails up. If you toss a bunch of pennies into the air, the chance that any particular one of them will be tails up is still 50%. If a particular penny lands heads up on the first toss, the chance that it will land tails up on the second toss is still 50%. That's exactly the model we need for radioactive decay since the chance of any particular atom decaying in one second is unaffected by the fact that it did not decay a second ago.

The lab procedure to mimic radioactive decay is simple. Place 100 pennies in a box. Toss the pennies onto a table surface or the floor. Any pennies that land tails up are assumed to have "radioactively decayed" on this toss. Stack the tails-up pennies into a column on the far side of the table. Return the remaining heads up pennies to the box and toss them again onto the table surface. Remove the tails up pennies and stack them into a second column right beside the first column. Repeat the process until all pennies have landed tails up. If no pennies are tails up on a particular toss, leave the column corresponding to that toss empty.

It's best if you actually do this experiment "live" (tossing pennies is fun!). However, if you don't have 100 pennies handy, you can use this coin decayer application . Just follow the instructions on that page.

Once you have completed the data-taking, look at your columns of pennies. If you didn't use the coin decayer application, you should draw a graph that represents the number of pennies in each column vs. the toss number of the column. Repeat the entire experimental procedure again and draw a new graph for the height of the pennies. Compare the graphs. What shape appears? To get an even more detailed look at the shape, you might think about how to combine the data for your group with the data from other groups.

Now try finding which function best approximates the shape you see. For example, you can compare your graphs to that of a straight line, *y = mx + b*, with *y* being the number of pennies in a column and *x* being the toss number of the column. Obviously, you will have to see if you can find any values of *m* and *b* which fit your data. Compare your graphs to that of a straight line with the appropriate slope. Does it look like you can get a good fit? Try finding whether a quadratic or cubic equation fits better. What about the exponential and sine curves?

Scientists are often faced with the difficulty of fitting data coming from processes for which they do not have a quantitative prediction. Often, a functional form that fits the data can be found, but it may or may not give insight as to what causes the shape of the data. It's considerably easier to fit data if you start with at least a *qualitative* prediction of the relationship between your data points. To evaluate another way of fitting the data, try using the math you learned in the population growth problem. First, we figure out how to describe the problem. Consider the observations:

- The number of pennies always decreases when you toss them (or at best does not change).
- The number of pennies that "decay" on any given toss is proportional to the number of pennies you tossed -- you cannot lose more pennies than you tossed, but the more you toss, the more you tend to lose.
- The number of pennies we started with was 100.

Considering these observations, write down the change equation that describes the behavior listed. Your equation should show the number of pennies that will decay for any particular toss. Remember, we assume that the number of decays depends only on the number of pennies we have just before each toss. You'll see later how to make use of the specific number of pennies you started with. Once you have the equation, use Maple to solve it. What functional form do you get? Now try to find the values of the variables that give the best "fits" to your data for the pennies.

We can now extrapolate our observations by asking what would happen if we used dice instead of pennies, where a die is said to "decay" if it lands, say, with one dot showing up? In this case, we assume a fair die so that the probability of any particular number showing up is about 1/6 for each toss of a die. What do you predict the curve for an experiment of 100 dice would look like? What would you expect for the theoretical curve that fits the data taken in the experiment? Try it out! Not every group needs to assume that one dot facing up indicates decay. You are allowed to choose which number of dots facing up shows a decayed die.

The mathematics of radioactive decay is useful for many branches of science far removed from nuclear physics. One reason is that, in the late 1940's, Willard F. Libby discovered radiocarbon, a radioactive isotope of carbon with a half-life of approximately 5600 years. All living organisms take in carbon through their food supply. While living, the ratio of radiocarbon to nonradioactive carbon that makes up the organism stays constant, since the organism takes in a constant supply of both in its food. After it dies, however, it no longer takes in either form of carbon. The ratio of radiocarbon to nonradioactive carbon then decreases with time as the radiocarbon decays away. The ratio decreases exponentially with time, so a 5600-year-old organic object has about half the radiocarbon/carbon ratio as a living organic object of the same type today.

The technique of radiocarbon dating has been used to date objects as old as 50,000 years and has therefore been of enormous significance to archaeologists and anthropologists. There is a great deal of current activity in archeology centered on the question of when humans first arrived in the Western Hemisphere -- thought for some 50 years to be about 11,500 years ago, until older sites were recently discovered. While the question is still subject to debate, there is full agreement on radiocarbon dating as a tool for determining the age of organic artifacts.

Geologists also use radioactive decay to study the evolution of the earth. The decays of an isotope of uranium, with a half-life of 4.5 billion years, and of rubidium, with a half-life of 50 billion years, are used to determine the age of rocks found on the surface of the earth and the surface of the moon. The age and magnetic orientation of rocks on the seabed floor indicate the times of reversal of the earth's magnetic field. Lately, mass spectrometers have been used to get very accurate ratios of elements so that ages can be determined with great precision.