# Charcoal used to make prehistoric drawings on cave walls in France was analyzed. The results were 4 mg carbon-14 (parent isotope) and 60 mg nitrogen (daughter isotope). The half-life of carbon-14 is 5,730 years. How old are the cave drawings?

Sep 15, 2017

The cave drawings are 22920 years old.

#### Explanation:

$A = {A}_{o} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{h}}$, where $A$ is the final amount of radioactive anything, ${A}_{o}$ is the initial amount, $t$ is the elapsed time, and $h$ is the half-life of the radioactive material.

At the time of the drawing, there were 0 g of nitrogen 14 and 64 g of Carbon 14.

There are now 4 g.

Plugging everything in, we have:

$4 = 64 \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{5730}} \to \frac{1}{16} = {\left(\frac{1}{2}\right)}^{\frac{t}{5730}} \to$
${\log}_{\frac{1}{2}} \left(\frac{1}{16}\right) = {\log}_{\frac{1}{2}} {\left(\frac{1}{2}\right)}^{\frac{t}{5730}} \to 4 = \frac{t}{5730} \to 22920 = t$.

The cave drawings are 22920 years old.