326 CHAPTER 5 Exponential and Logarithmic Functions (b) Using this value of k, what is the relative risk if the concentration is 0.17 percent? (c) Using the same value of k, what concentration of alcohol corresponds to a relative risk of 100? (d) If the law asserts that anyone with a relative risk of having an accident of 5 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with a DUI? (e) Compare this situation with that of Example 10. If you were a lawmaker, which situation would you support? Give your reasons. 138. The Marriage Problem There is an infamous problem from mathematics that attempts to quantify the number of potential mates one should date before choosing one’s “true love.” The function ( ) = − L x x x ln represents the probability of finding the ideal mate after rejecting the first x proportion of potential mates. For example, if you reject the first = 20% 0.20 of individuals you date, the probability of finding the ideal mate is ( ) ≈ L 0.2 0.322. So, if you want the probability of finding the ideal mate to be greater than 0.332 and you are only willing to date up to 20 individuals, you should reject the first ( ) = 0.2 20 4 individuals before attempting to decide on the ideal mate. Presumably, you are using those first 4 individuals to help you decide which traits you value in a mate. (a) Determine and interpret ( ) L 0.1 . (b) Determine and interpret ( ) L 0.6 . (c) What is the domain of L? (d) Graph ( ) = L L x over the domain. (e) Judging on the basis of the approach suggested by the model, what is the value of x that maximizes L? What is the highest probability of finding the ideal mate? 139. Challenge Problem Solve: ( ) = x log log 1 6 2 140. Challenge Problem Solve: ( ) [ ] = x log log log 0 2 4 3 141. Challenge Problem Solve: = + + x log 9 1 x 3 2 3 2 Explaining Concepts 142. Is there any function of the form α = < < α y x , 0 1, that increases more slowly than a logarithmic function whose base is greater than 1? Explain. 143. In the definition of the logarithmic function, the base a is not allowed to equal 1. Why? 144. Critical Thinking In buying a new car, one consideration might be how well the price of the car holds up over time. Different makes of cars have different depreciation rates. One way to compute a depreciation rate for a car is given here. Suppose that the current prices of a certain automobile are as shown in the table. Age inYears New 1 2 3 4 5 $38,000 $36,600 $32,400 $28,750 $25,400 $21,200 Use the formula ( ) = e New Old Rt to find R, the annual depreciation rate, for a specific time t. When might be the best time to trade in the car? Consult the NADA (“blue”) book and compare two like models that you are interested in. Which has the better depreciation rate? 150. Find an equation of the line that contains the points ( ) 0, 1 and ( ) − 8, 4. Write the equation in general form. 151. Solve: + = x2 17 45 152. Forensic Science The relationship between the height H of an adult female and the length x of the female’s tibia, in centimeters, is estimated by the linear model ( ) = + H x x 2.90 61.53. If incomplete skeletal remains of an adult female include a tibia measuring 30.9 centimeters, estimate the height of the female. Round to the nearest tenth. 153. For ( ) = f x x ,3 find ( ) ( ) − − f x f x 2 2 . 154. Factor completely: ( ) ( ) ( ) ( ) + ⋅ − + − ⋅ + x x x x 5 7 3 3 4 5 4 6 7 3 Retain Your Knowledge Problems 145–154 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 145. Find the real zeros of ( ) = − + g x x x 4 37 9. 4 2 What are the x-intercepts of the graph of g? 146. Find the average rate of change of ( ) = f x 9x from 1 2 to 1. 147. Use the Intermediate Value Theorem to show that the function ( ) = − − f x x x 4 2 7 3 2 has a real zero in the interval [ ] 1, 2 . 148. A complex polynomial function f of degree 4 with real coefficients has the zeros −1, 2, and −i 3 . Find the remaining zero(s) of f. Then find a polynomial function that has the zeros. 149. Solve: − − = x x 2 7 1 0 2 ‘Are You Prepared?’ Answers 1. (a) ≤ x 3 (b) <− x 2 or > x 3 2. <− x 4 or > x 1
RkJQdWJsaXNoZXIy NjM5ODQ=