Question

Carbon Dating: All living organisms contain two isotopes (types) of carbon; carbon-12 ( 12C) and carbon-14 (14C). Carbon-12 is stable, meaning that its atoms do not decay. Carbon-14 however, is radioactive or unstable, meaning that an atom of carbon-14 may decay. It does this by throwing off an electron and thus transforming itself into Nitrogen 14. The ratio of carbon-14 to carbon-12 is the same for all living organisims, about 1/10, 000. However, once an organisim dies the carbon-14 begins to decay while the carbon-12 remains, thus lowering the carbon-14 to carbon-12 ratio. The half-life or carbon-14 is about 5700 years, meaning that a carbon-14 atom has a 50% chance of decaying over a 5700 year span. In other words, after 5700 years about half of the carbon14 you started with will be gone. This means that a sample of 14C decays at a rate that is proportional to the amount currently present. As you learned in section 6.5 of your text, and as discussed in class, this means that if the amount present at time t is given by Q(t), then Q(t) satisfies the differential equation Q 0 (t) = k Q(t).

(1) You also learned that the solutions to this equation have the form Q(t) = Q(0)e kt . (2) Suppose that your team has dug up some kind of animal bone, and you want to figure out how long ago that animal died. After analyzing the bone you determine the following:

1. The bone weighs 900 grams. 2. The bone is 65% 12C. 3. The ratio of 14C to 12C is 1/54, 000. Answer the following questions. Recall that to receive credit your answer must be supported by appropriate work. There should be no “number only” answers. At the very least, show the formula you are using to obtain your answer. 1. How many grams of 12C are present in the bone?

2. How many grams of 14C are present in the bone?

3. How many grams of 14C would have been present in the bone when the animal died?

4. How many grams of 14C would have been present in the bone 5700 years after the animal died?

5. Determine the constant k in equation (2) above as it pertains to this problem.

6. Write down a formula for Q(t), where Q(t) is the amount of 14C present in the bone t years after the animal died. Your formula should contain only two variables; The dependent variable Q and the independent variable t.

7. Using your formula, calculate the amount of 14C present in the bone 10,000 years after the animal died.

8. Based on your findings, is the bone more than 10,000 years old?

9. Using your formula, calculate the amount of 14C present in the bone 20,000 years after the animal died.

10. Based on your findings, is the bone more than 20,000 years old?

11. Calculate the age of the bone algebraically using your formula. Show all work.

Answer 1