396 CHAPTER 6 Algebra, Graphs, and Functions In Example 9, we used a scientific calculator to evaluate − (2) . 7000/5600 The steps need to do this are described in the following Technology Tip. MATHEMATICS TODAY Determining the Age of Bones and Fossils Carbon-14 dating is a method by which scientists estimate the age of organic materials such as bones, cloth, wood, and plant fibers. This dating technique was discovered by Willard Frank Libby and his colleagues in 1949. In 1960, Libby received the Nobel Prize in Chemistry for his work with carbon-14 dating. Carbon-14 dating is reliable for dating objects up to approximately 60,000 years old. Why This Is Important Carbon-14 dating is used to identify the age of various items. It is invaluable to archaeologists and criminologists and is used in many other professions. Example 9 Carbon Dating Carbon dating is used by scientists to estimate the age of fossils, bones, and other items. The formula used in carbon dating is = − P t P ( ) 2 , t 0 /5600 where P 0 represents the original amount of carbon-14(C ) 14 present in an item and P t( ) represents the amount of C14 present after t years. An archeologist finds a bone that is estimated to be about 7000 years old. If the bone originally had 10 milligrams of C14 present, how many milligrams of C14 are present now? Solution In this example, = P 10 0 and = t 7000. Substituting these values into the formula gives = = ⋅ = ⋅ ≈ ≈ − − − P t P P ( ) 2 (7000) 10 2 10 2 10(0.4204482076) 4.204482076 t 0 /5600 7000/5600 1.25 Thus, the bone has about 4.20 milligrams of C14 present. 7 Now try Exercise 79 TECHNOLOGY TIP Evaluating an Exponential Function The steps to evaluate (2) 7000/5600 − on a calculator with a ∧ key are given below. ∧ − ÷ = 2 ( ( ) 7000 5600 ) .4204482076 Notice that before the 7000, we used the negative key −( ) , and not the − subtraction key. After the = key is pressed, the calculator displays the answer 0.4204482076. To evaluate 10(2) 7000/5600 − on a scientific calculator, we can press the following keys. × ∧ − ÷ = 10 2 ( ( ) 7000 5600 ) 4.204482076. What does the graph of an exponential function of the form = > ≠ y a a a , 0, 1, x look like? Examples 10 and 11 illustrate graphs of exponential functions. Example 10 Graphing an Exponential Function with a Base Greater Than 1 a) Graph = y 2 .x b) Determine the domain and range of the function. Solution a) We begin by substituting values for x and calculating the corresponding values for y. We then plot the ordered pairs and use these points to graph the function. The graph is shown in Fig. 6.49. Phaitoon/123RF
RkJQdWJsaXNoZXIy NjM5ODQ=