1. example 3 Carbon-14 Dating
Radioactive carbon-14 decays according to the equation
where y0 is the original amount and y is the amount of carbon-14 at time t years. To find
the age of a fossil if the original amount of carbon-14 was 1000 grams and the present
amount is 1 gram, we solve the equation
Find the age of the fossil analytically by converting the equation to logarithmic form.
Find the age of the fossil using graphical methods.
Chapter 5.3
2009 PBLPathways

2. Radioactive carbon-14 decays according to the equation
where y0 is the original amount and y is the amount of carbon-14 at time t years. To find
the age of a fossil if the original amount of carbon-14 was 1000 grams and the present
amount is 1 gram, we solve the equation
Find the age of the fossil analytically by converting the equation to logarithmic form.
Find the age of the fossil using graphical methods.

3. Find the age of the fossil analytically by converting the equation to logarithmic form.
To convert this equation to logarithmic form,

4. Find the age of the fossil analytically by converting the equation to logarithmic form.
we need to isolate the exponential function

5. Find the age of the fossil analytically by converting the equation to logarithmic form.
by dividing both sides by 1000.

6. Find the age of the fossil analytically by converting the equation to logarithmic form.
We get one-thousandth equal to e raised to the t.

7. Find the age of the fossil analytically by converting the equation to logarithmic form.
The output from the exponential form becomes the input to the natural logarithm.

8. Find the age of the fossil analytically by converting the equation to logarithmic form.
We get t by itself by dividing both sides by

9. Find the age of the fossil analytically by converting the equation to logarithmic form.
This gives t equal to ln(0.001) divided by

10. Find the age of the fossil using graphical methods.
Let’s check this solution graphically.

11. Find the age of the fossil using graphical methods.
To use the x-intercept method, we need to have a 0 on one side of the equation.

12. Find the age of the fossil using graphical methods.
0 equal to 1000 times e raised to the quantity t minus 1.

13. Find the age of the fossil using graphical methods.
Graph the right hand side of the equation in a window that includes the solution.

14. Find the age of the fossil using graphical methods.
In this case, a window of [0, 80,000] by [-3, 3] shows the x-intercept.

15. Find the age of the fossil using graphical methods.
( , 0)
The x-intercept is at about (57, , 0) which matches the solution from part a.