Chapter 6 | Applications of Integration
741
Just as systems exhibiting exponential growth have a constant doubling time, systems exhibiting exponential decay have a constant half-life. To calculate the half-life, we want to know when the quantity reaches half its original size. Therefore, we have y 0 2 = y 0 e − kt 1 2 = e − kt −ln2 = − kt t = ln2 k . Note : This is the same expression we came up with for doubling time. Definition If a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by Half-life = ln 2 k .
Example 6.46 Radiocarbon Dating
One of the most common applications of an exponential decay model is carbon dating. Carbon-14 decays (emits a radioactive particle) at a regular and consistent exponential rate. Therefore, if we know how much carbon was originally present in an object and how much carbon remains, we can determine the age of the object. The half- lifeof carbon-14 is approximately 5730 years—meaning, after that many years, half the material has converted from the original carbon-14 to the new nonradioactive nitrogen-14. Ifwe have 100 g carbon-14 today, how much is left in 50 years? If an artifact that originally contained 100 g of carbon now contains 10 g of carbon, how old is it? Round the answer to the nearest hundred years.
Solution We have
5730 = ln2 k k = ln2 5730 .
So, the model says
y =100 e −(ln 2/5730) t .
In 50 years, we have
y = 100 e −(ln 2/5730)(50) ≈ 99.40.
Therefore, in 50 years, 99.40 g of carbon-14 remains. To determine the age of the artifact, we must solve
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