Many scientists believe that there have been four Ice Ages in

the past 1 million years. Before the technique of carbon dating was known,
geologists erroneously believed that the retreat of the Fourth Ice Age began
about 25,000 years ago. In 1950, logs from ancient spruce trees ere found
under glacial debris near Two Creeks, WI. Geologists determined that these
trees had been crushed by the advance of ice during the Fourth Ice Age.
Wood from the spruce trees contained 27% of the level of 14C found in living
trees. Approximately how long did the Fourth Ice Age actually occur? Recall
that the decay constant of 14C is 0:00012.

That's a lot of words to ask the question:

What is t, if
exp(-.00012t) = 0.27 ?

Solve for t (in years)

To determine the duration of the Fourth Ice Age, we can use the concept of carbon dating and the information provided about the levels of 14C (carbon-14) in the ancient spruce trees compared to living trees.

First, let's understand the basics of carbon dating. Carbon-14 is a radioactive isotope of carbon that is present in the Earth's atmosphere. It is absorbed by living organisms, including trees, during photosynthesis. When an organism dies, it no longer absorbs carbon-14, and the existing carbon-14 begins to decay at a known rate.

The decay of carbon-14 follows an exponential decay model, described by the decay constant, which is given as 0.00012 in the question.

Now, let's calculate the age of the ancient spruce trees during the Fourth Ice Age:

1. Calculate the number of half-lives of carbon-14 in the wood:
- The ratio of carbon-14 in the ancient wood to that in living trees is 27%.
- This means the ancient wood has experienced (100% - 27%) = 73% decay.
- We can calculate the number of half-lives using the formula:
Number of half-lives = (ln(Ratio of remaining carbon-14) / ln(0.5))
Number of half-lives ≈ (ln(0.73) / ln(0.5))
Number of half-lives ≈ 2.544

2. Determine the age of the ancient wood based on the number of half-lives:
- One half-life of carbon-14 is the time taken for half of the carbon-14 to decay.
- The decay constant is given as 0.00012 (per year).
- The equation for determining the age of the ancient wood is:
Age = (Number of half-lives) / (Decay constant)

Age ≈ 2.544 / 0.00012
Age ≈ 21200 years

Therefore, based on the carbon dating of the ancient spruce trees, the Fourth Ice Age actually occurred for approximately 21,200 years.