358 CHAPTER 5 Exponential and Logarithmic Functions Estimating the Age of Ancient Tools Traces of burned wood along with ancient stone tools in an archeological dig in Chile were found to contain approximately 1.67% of the original amount of carbon-14. (a) If the half-life of carbon-14 is 5730 years, approximately when was the tree cut and burned? (b) Using a graphing utility, graph the relation between the percentage of carbon-14 remaining and time. (c) Use a graphing utility to determine the time that elapses until half of the carbon-14 remains. This answer should equal the half-life of carbon-14. (d) Use a graphing utility to verify the answer found in part (a). Solution EXAMPLE 3 (a) Using formula (3), the amount A of carbon-14 present at time t is ( ) = A t A ekt 0 where A0 is the original amount of carbon-14 present and k is a negative number. We first seek the number k. To find it, we use the fact that after 5730 years half of the original amount of carbon-14 remains, so ( ) = A A 5730 1 2 .0 Then Figure 56 = − Y e x 1 0.000120968 1 0 0 40,000 = = = = ≈ − ⋅ A A e e k k 1 2 1 2 5730 ln 1 2 1 5730 ln 1 2 0.000120968 k k 0 0 5730 5730 Divide both sides of the equation by A . 0 Rewrite as a logarithm. Formula (3), therefore, becomes ( ) = − A t A e t 0 0.000120968 If the amount A of carbon-14 now present is 1.67% of the original amount, it follows that = = − = = − ≈ − − A A e e t 0.0167 0.0167 0.000120968t ln0.0167 ln0.0167 0.000120968 33,830 years t t 0 0 0.000120968 0.000120968 Divide both sides of the equation by A . 0 Rewrite as a logarithm. The tree was cut and burned about 33,830 years ago. Archeologists may use this conclusion to argue that humans lived in the Americas nearly 34,000 years ago. (b) Figure 56 shows the graph of = − y e x 0.000120968 on a TI-84 Plus CE, where y is the fraction of carbon-14 present and x is the time. (c) Graph = − Y e x 1 0.000120968 and = Y 0.5, 2 where x is time. Use INTERSECT to find that it takes 5730 years until half the carbon-14 remains. The half-life of carbon-14 is 5730 years. (d) Graph = − Y e x 1 0.000120968 and = Y 0.0167 2 where x is time. Use INTERSECT to find that it takes 33,830 years until 1.67% of the carbon-14 remains. Now Work PROBLEM 3
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