By Request: Radiometric Dating

I was asked to explain the theory behind radiometric dating, one of the many scientific applications of calculus.

Radiometric dating seeks to determine the age of a material based on the relative amounts of certain isotopes in the material. These isotopes are chosen for various reasons. For example, a common type of radiometric dating is carbon-14 dating. This dating is used to determine the age of fossils. While an animal is living, it is exchanging matter with the environment and over time will have a level of carbon-14 equal to that of the environment. When the animal dies, it ceases to take in any more carbon-14. The carbon-14 will slowly decay to carbon-12 through beta decay. By comparing the amount of carbon-14 in the dead animal to the environmental amount, we can determine when the animal died.

But how can we do that?

The decay of a father isotope into a daughter isotope happens due to forces between subatomic particles within atoms. For example, the weak force, one of the four fundamental forces, is responsible for beta decay, the mechanism behind carbon-14 decay. Because decay depends on quantum mechanics, the type and rate of decay depends solely on the father isotope in question.

Since the probability of decay depends on the father isotope, we can form an equation to predict decay to the daughter isotope over time. How do we do this? If all the atoms we have are the same father isotope, then each has an equal probability of decaying. Therefore, the rate of decrease in the number of atoms that are the father isotope should be proportional to the number of atoms of the father isotope. Let us call the constant of proportionality lambda, and write the equation and solve it.


Click here for equation and solution


So we see that the number of atoms of the father isotope decays exponentially.

Challenge: You might have heard of half-life before. Can you express the half-life in terms of lambda?

By Request: Entrance Exam Problem

Sameer Hemmady in Albuquerque writes:

A dog is chasing a cat with uniform velocity “v” so that at any given time the dog is “aimed” directly at the cat. The cat upon seeing the dog runs rectilinearly and uniformly with velocity “u” with u<v. At the initial moment when the cat sees the dog, the vectors v and u are perpendicular to each other, and the dog and cat are separated by a distance “d”. How soon will the dog catch up to the cat? (Problem in our engineering entrance exam in India)

Entrance exam problems require quite a bit of ingenuity to solve. This is something to be expected. They are not testing your ability to simply differentiate or integrate, but to create an approach to the problem. This ability is important if you plan to use calculus in your career (I actually solved a problem similar to this in my job, involving a jet and a ballistic interceptor). Math problems do not present themselves as equations for you to solve, but you must form the equations yourself from meaningful quantities in the word problem. Creating an approach to a problem takes quite a bit of trial and error. Therefore, when solving this problem I will consider different approaches.

Solving entrance exam problems is a science in its own right, and learning all there is about it would take several semester-long classes. However, even if you have never been exposed to entrance exam problems, and especially even if this is your first semester of calculus, I encourage you to look at the solution to the problem, and ask questions if you have any.

Click here for solution