Lesson 7

1. A colony of microbes doubles in population every 6 hours. Explain why we could say that the population grows by a factor of \(\sqrt[6]{2}\) every hour.
2. A bacteria population decreases by a factor of \(\frac{1}{2}\) every 4 hours. Explain why we could also say that the population decays by a factor of \(\sqrt[4]{\frac12}\) every hour.

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.

1. At a certain point in time, a fossil had 3 picograms (a trillionth of a gram) of carbon-14. Complete the table with the missing mass of carbon-14 and 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

2. A scientist uses the expression \((2.5) \boldcdot \left(\frac{1}{2}\right)^{\frac{t}{5,730}}\) to model the number of picograms of carbon-14 remaining in a different fossil \(t\) years after 20,000 BC.
   1. What do the 2.5, \(\frac{1}{2}\), and 5,730 mean in this situation?
   2. Would more or less than 0.1 picogram of carbon-14 remain in this fossil today? Explain how you know.

The half-life of carbon-14 is about 5,730 years.

1. Pythagoras lived between 600 BCE and 500 BCE. Explain why the age of a papyrus from the time of Pythagoras is about half of a carbon-14 half-life.
2. Someone claims they have a papyrus scroll written by Pythagoras. Testing shows the scroll has 85% of its original amount of carbon-14 remaining. Explain why the scroll is likely a fake.

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 \(10^{\text-9}\), 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 \(f\) gives the number of nanograms of sodium remaining after \(t\) years then \(\displaystyle f(t) = 4 \boldcdot \left(\frac{1}{2}\right)^{\frac{t}{3}}\)

The 4 represents the number of nanograms in the sample when it was first measured, while the \(\frac{1}{2}\) and 3 show that the amount of sodium is cut in half every 3 years, because if you increase \(t\) by 3, you increase the exponent by 1.

How much of the sodium remains after one year? Using the equation, we find \(f(1) = 4 \boldcdot \left(\frac{1}{2}\right)^{\frac{1}{3}}\). 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 \(\frac12\) each time. If we multiply 0.5 nanogram (the mass of sodium-22 9 years after first being measured) by \(\frac12\) 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.