I’m reading The Greatest Show on Earth and it talks about measuring time on a geological scale. Although not the point of this post, it’s fascinating, so here’s a quick explanation:

### How radioactive dating works

Matter is made of atoms. Atoms come in different flavors, which we call elements. The number of protons is fixed for any given element, and is equal to the number of electrons. So, for example, carbon atoms have 6 protons and electrons while nitrogen atoms have 7.

The number of neutrons, however, is not fixed. Carbon atoms can have different numbers of neutrons. These different variations are called “isotopes”. The most common three isotopes of carbon are carbon-12 (98.9%), carbon-13 (1.1%), and carbon-14 (<0.01%). These isotopes have 6, 7, and 8 neutrons respectively. The number associated with each isotope is the “mass number” which equals the number of protons + neutrons – electrons are so light they don’t contribute to the mass number.

Certain isotopes are unstable, by which I mean they spontaneously decay into something else, at a predictable rate. The predictability of the rate of decay is key.

The decay tends to happen in one of three ways:

• Neutron turns into a proton
• A proton turns into a neutron
• A stray neutron happens to hit a nucleus and knocks out one proton, taking its place.

For example, potassium-40 decays into argon-40. The favored measure of decay rate is called the “half-life”, which is the amount of time it takes for half of the potassium-40 atoms to decay into argon-40 atoms. For this particular pairing, the half-life is 1.26 billion years.

Why does this help? Well, if you happen to know that at some moment in time in the past, let’s call it time X, there was only potassium-40 and no argon-40, then by measuring the ratio of potassium-40 to argon-40 now, you can compute the amount of time that has passed since X. It’s important to note that this only works if you know that, at time X, there was no argon-40. In this case, igneous rocks are solidified from molten rock (magma or lava) and at the moment of solidification, the rock contains potassium-40 but no argon-40.

So let’s say you take an igneous rock and measure the ratio of potassium-40 to argon-40 and it’s 0.5. That means half the potassium-40 has decayed into argon-40. Since we know the half-life of potassium-40 is 1.26 billion years, that means the rock was formed about 1.26 billion years ago. What if the ratio was 0.25? That means the amount of potassium-40 has halfed twice, so it’s been 2.52 billion years since that particular rock was formed. Hopefully it’s clear how this generalizes.

### A treasure trove of math problems

And here we come to the point of this post. It seems to me that this real-world scenario can motivate a host of interesting math questions, across a surprisingly broad range of difficulty levels.

#### Elementary School

For elementary school, you can set up the ratios to form an integer number of half-lives:

• We start with 100 grams of potassium-40. How long will it be until we only have 25 grams of potassium-40?
• We currently have 25 grams of potassium-40 and 75 grams of argon-40. When did this rock form?

#### High School

For high school, you can make the ratios whatever you want, which requires logs.

• We start with 100 grams of potassium-40. How long until we have 32 grams of potassium-40?
• We currently have 32 grams of potassium-40 and 68 grams of argon-40. When did the rock form?

You can also get into “experiment design” or “thinking like a scientist”:

• You have an isotope and you don’t know the half-life. You start with 100g and after 1 year you have 98g. What’s the half-life?
• Say you have 1kg of rubidium-87, which has a half-life of 49 billion years. How long do you expect to wait until 1 atom decays into strontium-97?
• You have 100kg of rubidium-87 and you’re attempting to verify the half-life. Say your measuring device is only accurate enough to detect an element when there is at least 1 microgram of it. How long do you expect to have to wait in order to detect any strontium?
• You’re designing an experiment to verify the half-life of rubidium-87 and the longest you’d like to run your experiment is 1 year. You start with 1kg of rubidium-87. How sensitive does your measuring device need to be (i.e. what is the smallest weight of an element that you will need to be able to detect)?

#### College

For college, you can start to introduce differential equations.

• Isotope A decays into B with a half-life of 30 years. Come up with an equation for the ratio of A:B as a function of time t since there was only A.
• Isotope A decays into B with a half-life of 30 years and B decays into C with a half-life of 10 years. Set up a system of differential equations that represents this system.

You can make this really hard:

• Isotope A decays into B with a half-life or 30 years and B decays into C with a half-life of 10 years. You come across a rock with a ratio of A:B:C of #:#:#. When was there only A?
• Is it harder if there was never a time when there was only A? Maybe you specify that, at some point in the past the ratio was #:#:#. How much time has elapsed since then?
• Solve the system of differential equations to give a closed-form for how much of A, B, and C there are at any time t.

Is it harder if you make a loop?

• Isotope A decays into B with a half-life of 30 years. B -> C w/half-life of 10 years. And C -> A w/half-life of 15 years.