Math Exploration: Fossil Dating

Intro

From a young age I have found

fossils everywhere: In my backyard, the playground, and all over parks. The

idea of prehistoric creatures has always fascinated me, but I could never

understand how paleontologists know how old they are. I wondered how paleontologists

were able to know how fossil relates to evolution and how they use fossils to

classify eras. This brought about the question: How can math be used to

determine the age of fossils?

In order to

answer this question, one must determine the different methods of fossil dating

and what equations are involved in these types of dating. Exponential equations

and logarithms can be used to show how fossils age over time. Paleontologists

use these equations to classify eras and show the evolution of species.

Scientists

use several methods to date fossils, mainly involving mathematical formulas.

Radiometric dating is used most often to determine the absolute date of the

fossil. There are four types of radiometric dating which use mathematical

equations to determine the exact date of fossils: Potassium-Argon Dating,

Rubidium-Strontium Dating, and Uranium-Lead Dating.

Background

In order to understand the math

behind Fossil dating, an understanding of the terminology must be collected.

The different types of dating, Potassium-Argon, Rubidium-Strontium, and

Uranium-Lead, all require different techniques and equations that can determine

the age of fossils. These methods are used to determine the age of the rocks in

which the fossil is found, therefore, the scientists can determine when the

fossil lived. Understanding how each one works and the methods used to date

fossils gives one a better understanding of the equations associated with each

technique.

Potassium-Argon

dating is a technique used for older fossils, as it can date back farther than

many methods. The radioisotope, Potassium-40 is said to dissolve in Argon-40.

By figuring the rate of decay in Potassium-40 and finding the ratio of

Potassium to Argon, one can find the date of a fossil. (Earlham 1)

Rubidium-Strontium

dating is an accurate type of dating because it has such a high half-life. As

igneous rock cools, rubidium decays, turning into strontium, therefore creating

a ratio between the two. The ratio determines how old the rock and fossil are.

Uranium-Lead

dating is a more difficult type, as the two isotopes are not directly related.

Also, it is difficult to retain in rocks. In order for the isotope to become

lead, it must undergo several short transformations (Earlham 1). Despite its

difficulties, it can be a precise way to determine the age of rocks. (Kerr 789)

Aim

The

investigative nature of this work attempts to explain how paleontologists and

other scientists use mathematics to determine the exact and relative age of

fossils. By determining the different variables that can be manipulated, an

understanding of how dating works can be determined, depending on the various

types of radiometric dating. The collection of research will create an overall

better understanding of what radiometric dating is and how it is used.

Research

The following equation can be used

for each type of radiometric dating:

Figure 1 (Radiometric Time Scale, 1)

Each of the independent variables, D, P, ?,

determine what the dependent variable, t, will be. By plugging in each variable

to the set equation, one is able to find the age of the rock, and therefore the

fossil. For the sake of simplicity, in order to find the decay constant, the

bottom formula in Figure 1 will be rewritten as: . Depending on the method and presence

of particular isotopes, the equation will produce a specified number which

states the age of the rock.

The following table gives the

respected daughter products and current scientifically accepted half-lifes of

each isotope used in radiometric dating:

Figure 2 (Radiometric Time Scale, 1)

A “half-life of an isotope is defined as the amount of time

it takes for there to be half the initial amount of the radioactive isotope

present” (LeVarge 1). The respective half-lives will be used for the equation

to determine the appropriate decay constant.

Potassium-Argon

Dating

Potassium-Argon dating is an

effective method to date relatively old materials. Some dating, up to four

billion years old is obtainable. By comparing the ratio of Potassium-40 (K-40)

to Argon-40 (Ar-40) and knowing the

decay rate of K-40, the date of the material is found.

For example, if there are 1000 atoms of K-40

(the parent atom), and 300 atoms of Ar-40 (the daughter atom), the age is

obtainable by plugging in to the equation. In order to have all the variables

required, one must first find the appropriate decay constant (?). To find this,

using a graphing calculator, use the currently accepted half-life, 1.25 billion

years as and solve:

?=

?= 0.5545177444

For the sake of this example, the solutions will be rounded

to the hundredths place. Using this number, plug in all the variables into the

equation shown in Figure 1 to find the age. Use a graphing calculator for the

most accurate results.

Therefore, the age of the fossil is 0.48 billion

years old. Because the number used to determine ? was in billions of years, the

end result is also in billions of years. This process can be used in other

methods of dating as well, using the accepted half-lives and the ratio of

parent and daughter isotopes.

Rubidium-Strontium Dating

Strontium-87 is a stable isotope

that does not change due to decay. As Rubidium-87 decays, it transforms into

Strontium-87. Therefore, the only way age can be obtained using this method is

by comparing the amount of Rubidium-87 to Strontium-87.

In

an example similar to the one found in Potassium-Argon dating, a fossil has

4000 parent atoms and 200 daughter atoms. Repeating the same steps used in the

previous example; the half-life, 48.8 billion years is used to determine ?.

?=

?= 0.142038357

Using the ? value and the number of parent and

daughter atoms, the age can be found:

The fossil’s age is 0.35 billion years old. The same process

can be used for various types of radiometric dating because each method uses a

ratio to determine its age. As long as the variables that alter the equation

are relative to the type of dating, the results will be produced for that type

of dating.

Uranium-Lead Dating

Uranium is an unstable, reactive

element. It sheds nuclear particles until it results in lead. When a mineral

grain forms, it begins absorbing lead atoms that accumulate over time. By

determining the amount of lead atoms, the date can be determined.

Once

again, the steps from the first two examples can be used to determine the age

of a exampled fossil. Using 500 parent atoms of Uranium-238 and 2000 daughter

atoms of Lead-206, use the half-life, 4.5 billion years to find ?:

?=

?= 0.1540327068

Use the number of atoms for both the daughter and

parent, as well as ?=0.15 to find the age:

The fossil is 10.73 billion years old. This may seem higher

in age than the than the other examples using other methods of dating.

This is because in this example, the

daughter atoms were higher than the parent atoms. As the parent atoms decay to

become the daughter atoms the length of its life increases, as these processes

take a while. Therefore, the higher the number of daughter atoms in comparison

to parent atoms, the older the fossil will be.

Reflection

My

collection of work, with included examples shows how math applies to real life

situations. The exponential equations that I practice in class are able to be

manipulated and incorporated into an idea that I find interest in and I want to

understand. By completing this, I am able to understand the importance of math

in everyday life. While reflecting on the work of this paper, I find that I

could change several things. I had to trust the work of others, due to the lack

of supplies and accessibility that would help me conduct my own experiment.

Therefore, any variation in their works could impact mine. However, I have

successfully proven the use of mathematics in real terms and have a better

understanding, both on radiometric dating.

Conclusion

By manipulating the variables in an

exponential equation, it is possible to determine the age of a fossil quickly

and accurately. The different options available for radiometric dating provide

different results which can be used in different ways to support claims. Not

only is this an example of how exponents are used to solve real world problems,

but also how formulas is an effective and fast method to getting answers.

Bibliography

Alden,

Andrew. “Uranium-Lead Dating.” ThoughtCo,

www.thoughtco.com/uranium-lead-dating-1440810.

Earlham College – Geology 211 – Radiometric Dating,

legacy.earlham.edu/~smithal/radiometric-types.htm.

Kerr,

Richard A. “Stretching the Reign of Early Animals”. Science, May

2000, Vol. 288 Issue 5467,

page 789.

LeVarge,

Sheree. “Carbon Dating.” BioMath: Carbon

Dating,

www.biology.arizona.edu/biomath/tutorials/applications/carbon.html.

Potassium-Argon Dating,

www.anth.ucsb.edu/faculty/stsmith/classes/anth3/courseware/Chronology/09_Potassium_Argon_Dating.html.

Radioactive Dating, chem.tufts.edu/science/FrankSteiger/radioact.htm.

“RADIOMETRIC

TIME SCALE.” Geologic Time: Radiometric

Time Scale,

pubs.usgs.gov/gip/geotime/radiometric.html.

Appendix

Figure 1:

(Radiometric Time Scale, 1)

Figure 2:

(Radiometric Time Scale, 1)