James M. answered • 03/24/19

Math nerd interested in furthering math education

For these types of questions it is important to remember the basic exponential equation A=I*e^(kt) where A is the amount we have, I is the initial amount, e is about 2.718 (your calculator has the exact number), k is the constant of growth or decay, and t is the time elapsed.

In this specific question, the first thing you should do is to figure out what k is. You can do this by using the fact that the half life of carbon-14 is 5700 years. To find k set up your equation as follows:

.5=e^5700k

To figure out k here take the natural log of both sides:

ln(.5)=5700k

Then solve for k:

k=ln(.5)/5700 ~~ -1.216*10^-4

Now that we know what k is, we can go back to the original question and plug in everything we know! Our new equation should look like this:

.04=e^((ln(.5)/5700)t)

Solving for t works the same as solving for k, take the natural log of both sides:

ln(.04)=(ln(.5)/5700)t

Then solve for t:

t=ln(.04)/(ln(.5)/5700) ~~ 26469.98 years