Unit 5, Lesson 7
Interpreting and Using Exponential Functions
Algebra 2
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Carbon-14 is used to find the age of certain artifacts and fossils. It has a half-life of 5,730 years, so if an object has carbon-14, it loses half of it every 5,730 years.
| number of years after fossil had 3 picograms of carbon-14 |
mass of carbon-14 in picograms |
|---|---|
| 0 | 3 |
| 1,910 | |
| 5,730 | |
| 0.75 |
The half-life of carbon-14 is about 5,730 years.
Some substances change over time through a process called radioactive decay, and their rate of decay can be measured or estimated. Let’s take sodium-22 as an example.
Suppose a scientist finds 4 nanograms of sodium-22 in a sample of an artifact. (One nanogram is 1 billionth, or , of a gram.) Approximately every 3 years, half of the sodium-22 decays. We can represent this change with a table.
| number of years after first being measured |
mass of sodium-22 in nanograms |
|---|---|
| 0 | 4 |
| 3 | 2 |
| 6 | 1 |
| 9 | 0.5 |
This can also be represented by an equation. If the function gives the number of nanograms of sodium-22 remaining after years, then
The 4 represents the number of nanograms in the sample when it was first measured, while the and 3 show that the amount of sodium-22 is cut in half every 3 years, because if you increase by 3, you increase the exponent by 1.
How much of the sodium-22 remains after one year? Using the equation, we find . This is about 3.2 nanograms.
About how many years after the first measurement will there be about 0.015 nanogram of sodium-22? One way to find out is by extending the table and multiplying the mass of sodium-22 by each time. If we multiply 0.5 nanogram (the mass of sodium-22 9 years after first being measured) by five more times, the mass is about 0.016 nanogram. For sodium-22, five half-lives means 15 years, so 24 years after the initial measurement, the amount of sodium-22 will be about 0.015 nanogram.
Archaeologists and scientists use exponential functions to help estimate the ages of ancient things.