Exercises 2.4 Practice Exercises 2.4 1. In 1980 the U.S. population was about 225 million, and in 1990 it was about 250

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Exercises 2 4 Practice Exercises 2 4 1 In 1980 The U S Population Was About 225 Million And In 1990 It Was About 250 1 (42.26 KiB) Viewed 42 times

Exercises 2 4 Practice Exercises 2 4 1 In 1980 The U S Population Was About 225 Million And In 1990 It Was About 250 2 (62.28 KiB) Viewed 42 times

Exercises 2.4 Practice Exercises 2.4 1. In 1980 the U.S. population was about 225 million, and in 1990 it was about 250 million. Assuming that the pop- ulation is modeled by an exponential function yo where y is the population in millions and r is the number of years after 1980, determine yo and k. Also, predict the population in 2020. 11. 2.4 Natural Logarithm and Applications Dividing both sides by yo and taking logarithms give in (/) Solving for r gives In Exercises 1-8, rewrite the given expression as an ex ponential with base e in 5. a 6.3 a/s In Exercises 9-16, evaluate the given logarithm as a num ber in decimal form. Do not use a calculator 10. In(e) 12. In(e)) 14. 13. In 4. √ 8. 3 The decay of carbon-14 is the basis of the method of carbon dating, invented in the 1940s by Willard Libby, an American chemist and Nobel Laureate. We will come back to that method in Chapter 4. 18. In(√2) 21. In(e) (15) 16.¹2 In Exercises 17-22, use the approximation In 20.693 to approximate the given quantity. Do not use a calcula- for 17 In (2e) 20. In(e-) In 2 5,730 19 In 22. In 81 5.730 In(0.85) In 2 -In(0.85). 1,343.5 years. 2. Plutonium-241 decays according to the formula y where t is measured in years. Find its half- - 26. life. 3. If you put money in an account that pays 10% interest, compounded continuously, how long will it take for your money to quadruple? 153 In Exercises 23-34, simplify the given expression as much as possible. 23./2 27. In 28. (39, In(x+1)=1 41. In(x)=5 43.8 (45) In(2x-2)=0 (29) Inte) 30. 32. In(e)-1 33. 34 Inte)- In Exercises 35-45, solve the given equation for s. Do not use a calculator. Your answer may involve natural logarithms or the number e. (35)²+²=7 37.²21 31.- 36.¹=7 38. = 10 40. In(2x)=3 42. In(2x-3)=-1 44.) -8 46. In(Inx)=0

154 47-52 calor to find correct to at let even decimal place wheat asing the equation ormace function Then we the equation over check your anwe 47, In (3x + 5) - 2.47 49.-8.371 Chapter 2 EXPONENTIALS AND LOGARITHAS 51.-31 52.6 In Exercises 33-36, write the given expression as a quo dent of natural logarithms Then use a calculator to find a decimal approximation 53 Jog, 12 (85) log, 10 48 In(1-¹)+00318-0 50. 2+In(4)=0. 54. Jog 5 56 log; 57. Use formula (24) to show that if a and bare positive num- bers, 1 log, b log, a (Hint: Write both sides as quotients of natural logarithms.). In Exercises 58-61, sketch the graph of the given func tion by referring to the graph of y Inx (shown in Fig we 24.11 58 y=In(1+1) (-) 60. y In 59. y=1+Inx 61. y In -In (²) In Exercises 62-65, match the equation to one of the graphs in Figure 2.4.4 62. y=In(√) 64. yIn(3x) 66. Ir 54,000 is deposited in an account paying 6% interest per year, compounded continuously, how long will it take for the balance to reach $6,000? 63. yIn(3) (-) 65. y = In 68 Suppose you invest $5,000 in an account that pays interest. compounded continuously. (a) How much money is in the account after 8 years if the annual interest rate is 4%? (b) If you want the account to contain $8,000 after 8 years, what annual interest rate is needed? kr. A (4) 69. In an experiment on learning patterns, the psychologists Miller and Dollard recorded the time it took a 6-year-old girl to find a hidden piece of candy in a series of tries. It took her 210 seconds to find the candy on the first try. and it took 86 seconds on the second try. Assume that the Figure 2.4-4 (d) amount of time it took is modeled by T = Ae, where n is the number of tries and & is a constant. (a) Find the constants A and k (b) If the model is correct, how much time should it take the girl on the ninth try? (The actual experimental result was 2 seconds) Source: N. Miller and J. Dollard, Social Learning and Imitation, Yale University Press, 1941. 70. A crime scene investigator knows that h hours after death, the human body has a temperature of 67. If you put money in an account that pays 7.5% interest. compounded continuously, how long will it take for your 71. A colony of insects had a population of 4,000 when first money to triple? observed, and 5 days later it had grown to 6,000. Assume the population is growing exponentially. (a) Find a formula of the form y = Ae", where y is the population and r is the number of days after the initial ob- servation. Use a calculator to determine k to four decimal places. T=T +(98.6-7)(0.6)*. the air surrounding the body. Find the time of death of a where T, is the temperature (in degrees Fahrenheit) of person found in a room of 72°F constant air temperature, when its body temperature is 78°F. (b) According to this formula, what will the population be 8 days after the initial observation? (c) According to this formula, how many days after the initial observation will the population reach 12,000? 72. In 1940 the U.S. population was 131.67 million, and in 2000 it was 281 million.