Half life question

Question

Analysis indicated that a scroll wrapping contained 76% of the original carbon-14. The half life of carbon-14 is 5715 years. Set up & solve equation to find age

Answer 1

Using half-life is an example of "exponential decay."  The rule for using half-life is:

 

A = I * 0.5^t/H

 

I is the initial amount of the substance

t is the amount of time that has passed

H is the half-life, in the same units as t.  In this case, its 5715 years

A is the amount of the substance remaining after t has passed

 

In this case, they don't give you the initial amount of carbon-14 or the final amount, but they do give you the percentage of the original carbon-14 remaining.  So you can take the equation above and divide both sides by I:

 

A/I = 0.5^t/H

 

A/I is the ratio of the final amount to the original amount, or the fraction of the original amount that's remaining.  Well, that's a number they DO give you -- 76%, or 0.76

 

So now you have the following equation:

 

0.76 = 0.5^t/5715

 

Take the natural log of both sides:

ln (0.76) = t / 5715 * ln (0.5)

 

Solve for t:

5715 * ln (0.76) / ln (0.5) = t

 

Work this out on a calculator, and you get:

 

t = 2262.7

 

So the age of the sample is 2262.7 years