Name:______Period:______Date:______
Algebra 2 Unit 2 Modeling with Functions Performance Task 4
Excavating Exponential Decay
Congratulations, you have just completed one of the most exciting digs of your archeological career. You have discovered several fossils near the DiamondValleyLakein Hemet, California. Scientists throughout the world are anxiously awaiting your mathematical calculations determining the ages of the fossils you have discovered. Once again, congratulations on this remarkable discovery! Be careful that you calculate the age of the fossils correctly, you wouldn’t want a mathematical error to put a blemish on such a great archeological discovery.
Fossil #1 / Fossil #2 / Fossil #3 / Fossil #4Carbon 14 Data Analysis:
25% remaining / Carbon 14 Data Analysis:
20% remaining / Carbon 14 Data Analysis:
38% remaining / Carbon 14 Data Analysis:
22% remaining
How old is Fossil #1? / How old is Fossil #2? / How old is Fossil #3? / How old is Fossil #4?
It your job to present your findings to the scientists of the world. Your presentation must include a paragraph stating the ages of each of the fossils you discovered. Your paragraph should include the following information:
(1) The amount of Carbon 14 that was present at the time the fossils were tested.
(2) The mathematics you used to determine the age of each of the fossils.
(3) Which fossil you are the most excited about and why.
*Remember, your report may be seen by top scientists throughout the world. Make sure you present a well-written, neat, accurate, and informative report regarding your exciting archeological discovery.
ADVANCED / PROFICIENT / Progressing / Beginning☐Correctly calculate the age of a fossil with 60% remaining
☐Correctly calculate the age of a fossil with 10% remaining
☐Correctly plot the data points of all 6 fossils.
☐Use the graph to fill in the table on question #5 on ADVACED sheet. / ☐Correctly calculate the value of k.
☐Correctly calculated the age of Fossil #1
☐Correctly calculated the age of Fossil #2
☐Correctly calculated the age of Fossil #3
☐Correctly calculated the age of Fossil #4
☐All steps are shown
☐Paragraph is well written and address the fossil that you are the most excited about / ☐Meets 4-5 of the “Proficient” requirements / ☐Meets fewer than 4 of the “Proficient” Requirements
Excavating Exponential Decay Supplemental Student Sheet
Information on Carbon Dating: (Please read the paragraphs below.)
Archaeologists use the exponential, radioactive decay of carbon 14 to estimate the death dates of organic material. The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into carbon 12 and other particles. Carbon is naturally in all living organisms and is replenished in the tissues by eating other organisms or by breathing air that contains carbon. At any particular time all living organisms have approximately the same ratio of carbon 12 to carbon 14 in their tissues. When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to carbon 12 changes the ratio of carbon 12 to carbon 14. Experts can compare the ratio of carbon 12 to carbon 14 in dead material to the ratio when the organism was alive to estimate the date of its death. Radiocarbon dating can be used on samples of bone, cloth, wood and plant fibers.
The half-life of a radioactive isotope describes the amount of time that it takes half of the isotope in a sample to decay. In the case of radiocarbon dating, the half-life of carbon 14 is 5,730 years. This half life is a relatively small number, which means that carbon 14 dating is not particularly helpful for very recent deaths and deaths more than 50,000 years ago. After 5,730 years, the amount of carbon 14 left in the body is half of the original amount. If the amount of carbon 14 is halved every 5,730 years, it will not take very long to reach an amount that is too small to analyze. When finding the age of an organic organism we need to consider the half-life of carbon 14.
The formula for carbon 14-dating is , where = Initial Amount, = Amount remaining, k = constant, t = time.
Before we can use the formula, we must find k based on our problem. Use the following questions and your knowledge of natural logs to help you calculate k.
1) If you start with 100% of an object, what amount would remain in its half life? ______
2) What if instead of 100%, you start with 1 whole object. What amount would remain in its half life? ______
3) If 1 whole object is your initial amount and your answer to #2 is your amount remaining, how would you fill in the following formula?
4) Now go back through the reading above, how many years represent the half life of an object? ______
5) Use the formula you started in #3 and the answer to #4 to fill the time into the formula.
6) Now use your knowledge of natural logs to solve the equation you found in #5 for k. Show your steps and write your answer below. Keep your answer accurate to 9 decimal places.
Now that you have found k, use it to calculate the age of each fossil. Hint: when you plug in your amount remaining convert it to decimal point first and use 1 as your initial amount.
Name:______Period:______Date:______
Algebra 2 Unit 2 Modeling with Functions Performance Task 4
ADVANCED
1) Using the same formula,from the rest of the task calculate the age of a fossil with 60% remaining.
2) Using the same formula, from the rest of the task calculate the age of a fossil with 10% remaining.
3) Create a table of values from the 6 data points you have calculated.
Age of Fossil / Percent Remaining4) Now plot these points on the graph on the back and sketch a curve. Hint: it should look like an exponential function.
5) Use your graph to estimate how much of the fossil will remain at each data value in the table below.
Age of Fossil / Percent Remaining6500
10,000
15,300