# How to solve a carbon dating problem

Since we are looking for the age of the paintings, what variable are we looking for? It looks like we don’t have any values to plug into A or Ao.However, the **problem** did say that the paintings that were found contained 20% of the original **carbon**-14.Examples of exponential decay are radioactive decay and population decrease.

Some examples of exponential growth are population growth and financial growth.

The information found, can help predict what a population for a city or colony would be in the future or what the value of your house is in ten years.

We will still be using the same formula we did to answer the questions above, we will just be using it to find a different variable.

Plugging in 60 for A and solving for t we get: Since we are looking for when, what variable do we need to find? What are we going to plug in for A in this **problem**? Plugging 200000 for A in the model we get: t is the amount of time that has past.

As mentioned above, in the general growth formula, k is a constant that represents the growth rate. Since we are looking for the population, what variable are we finding? What are we going to plug in for t in this *problem*?

Our initial year is 1994, and since t represents years after 1994, we can get t from 2005 - 1994, which would be 11.The reason I showed you using the formula was to get you use to it.Just note that when it is the initial year, t is 0, so you will have e raised to the 0 power which means it will simplify to be 1 and you are left with whatever Ao is. Well, k = .0198026, so converting that to percent we get 1.98026% for our answer.Plugging in 10000 for t and solving for A we get: : A certain radioactive isotope element decays exponentially according to the model , where A is the number of grams of the isotope at the end of t days and Ao is the number of grams present initially. If we are looking for the half-life of this isotope, what variable are we seeking? It looks like we don’t have any values to plug into A or Ao.However, the **problem** did say that we were interested in the HALF-life, which would mean ½ of the initial amount (Ao) would be present at the end (A) of that time. Replacing A with .5 Ao and solving for t we get: : Prehistoric cave paintings were discovered in a cave in Egypt. Using the exponential decay model for **carbon**-14, , estimate the age of the paintings.If you need a review on these topics, feel free to go to Tutorial 42: Exponential Functions and Tutorial 45: Exponential Equations. For example, if the model is set up at an initial year of 2000 and you need to find out what the value is in the year 2010, t would be 2010 - 2000 = 10 years.