Radiation counters are used to detect the electrons given off by decaying Carbon-14 as it turns into nitrogen.
In order to date the artifact, the amount of Carbon-14 is compared to the amount of Carbon-12 (the stable form of carbon) to determine how much radiocarbon has decayed.
This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon $14$, that is, as soon as it dies.
Carbon $14$ dating is a fascinating topic and much information can be found on Wikipedia.
Many factors limit the accuracy of using Carbon 14 for dating including This problem is intended for instructional purposes only.
It provides an interesting and important example of mathematical modeling with an exponential function. The 1/13 ratio gets the correct answer of 21,203 years.
Cosmic rays enter the earth's atmosphere in large numbers every day and when one collides with an atom in the atmosphere, it can create a secondary cosmic ray in the form of an energetic neutron.