6.10 Functions and Their Graphs 399 Example 13 World Population Growth The world population in 2021 was about 7.89 billion people. Assume that the world population continues to grow exponentially at the growth rate of about 0.9% per year. The expected world population, P t( ), in billions of people, t years after 2021, can be estimated by the formula = P t e ( ) 7.89 .t 0.009 Use this formula to estimate the world population in the year 2031. Source: Census.gov Solution Since 2031 is 10 years after 2021, we substitute = t 10 into the formula. = = = ≈ ≈ P t e P e e ( ) 7.89 (10) 7.89 7.89 7.89(1.094174284) 8.633035098 t 0.009 0.009(10) 0.09 Thus, in 2031, the world population is expected to be about 8.63 billion people. 7 Now try Exercise 83 Example 14 Evaluating an Exponential Decay Function Plutonium, a radioactive material used in most nuclear reactors, decays exponentially at a rate of 0.003% per year. If there are originally 2000 grams of plutonium, the amount of plutonium, P, remaining after t years is P t e ( ) 2000 .t 0.00003 = − How much plutonium will remain after 50 years? Solution Substitute 50 years for t in the function, then evaluate using a calculator as described earlier. P t P ( ) 2000e (50) 2000e 2000e 2000(0.99850112244) 1997.0 grams t 0.00003 0.00003(50) 0.0015 = = = ≈ ≈ − − − Thus, after 50 years, the amount of plutonium remaining will be about 1997 grams. 7 Now try Exercise 85 Instructor Resources for Section 6.10 in MyLab Math • Objective-Level Videos 6.10 • Animation: Linear, Quadratic and Exponential Growth • PowerPoint Lecture Slides 6.10 • MyLab Exercises and Assignments 6.10 • Chapter 6 Projects Recall that all exponential functions of the form = y ax or = f x a ( ) ,x where > a 1 will have the general shape of the graph illustrated in Fig. 6.49. Since the natural base, ≈ e 2.7183, is greater than 1, the shape of the graph of the exponential function, = y exor = f x e ( ) ,x will also be similar to the graph in Fig. 6.49. Exercises Warm Up Exercises In Exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. Any set of ordered pairs is called a(n) ________ . Relation 2. A special type of relation where each value of the independent variable corresponds to exactly one value of the dependent variable is called a(n) ________. Function 3. The set of values that can be used for the independent variable is called the ________ of the function. Domain SECTION 6.10
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