310 CHAPTER 5 Exponential and Logarithmic Functions 116. Spreading of Rumors A model for the number N of people in a college community who have heard a certain rumor is ( ) = − − N P e 1 d 0.15 where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 1000 students, how many students will have heard the rumor after 3 days? 117. Exponential Probability Between 12:00 pm and 1:00 pm, cars arrive at Citibank’s drive-thru at the rate of 6 cars per hour (0.1 car per minute). The following formula from probability can be used to determine the probability that a car arrives within t minutes of 12:00 pm. ( ) = − − F t e 1 t 0.1 (a) Determine the probability that a car arrives within 10 minutes of 12:00 pm (that is, before 12:10 pm). (b) Determine the probability that a car arrives within 40 minutes of 12:00 pm (before 12:40 pm). (c) What does F approach as t becomes unbounded in the positive direction? (d) Graph F using a graphing utility. (e) Using INTERSECT, determine how many minutes are needed for the probability to reach 50%. 118. Exponential Probability Between 5:00 pm and 6:00 pm, cars arrive at Jiffy Lube at the rate of 9 cars per hour (0.15 car per minute). This formula from probability can be used to determine the probability that a car arrives within t minutes of 5:00 pm: ( ) = − − F t e 1 t 0.15 (a) Determine the probability that a car arrives within 15 minutes of 5:00 pm (that is, before 5:15 pm). (b) Determine the probability that a car arrives within 30 minutes of 5:00 pm (before 5:30 pm). (c) What does F approach as t becomes unbounded in the positive direction? (d) Graph F. (e) Using INTERSECT, determine how many minutes are needed for the probability to reach 60%. 119. Poisson Probability Between 5:00 pm and 6:00 pm, cars arrive at a McDonald’s drive-thru at the rate of 20 cars per hour. The following formula from probability can be used to determine the probability that x cars arrive between 5:00 pm and 6:00 pm. P x e x 20 ! x 20 ( ) = − where ( ) ( ) = ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅ x x x x ! 1 2 3 2 1 (a) Determine the probability that = x 15 cars arrive between 5:00 pm and 6:00 pm. (b) Determine the probability that = x 20 cars arrive between 5:00 pm and 6:00 pm. 120. Poisson Probability People enter a line for the Demon Roller Coaster at the rate of 4 per minute. The following formula from probability can be used to determine the probability that x people arrive within the next minute. P x e x 4 ! x 4 ( ) = − where ( ) ( ) = ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅ x x x x ! 1 2 3 2 1 110. Atmospheric Pressure The atmospheric pressure p on a balloon or airplane decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height h (in kilometers) above sea level by the function ( ) = − p h e 760 h 0.145 (a) Find the atmospheric pressure at a height of 2 km (over a mile). (b) What is it at a height of 10 kilometers (over 30,000 feet)? 111. Depreciation The price p, in dollars, of a Honda Civic EX-L sedan that is x years old is modeled by ( ) = ⋅ p x 25,495 0.90x (a) How much should a 3-year-old Civic EX-L sedan cost? (b) How much should a 9-year-old Civic EX-L sedan cost? (c) Explain the meaning of the base 0.90 in this problem. 112. Healing of Wounds The normal healing of wounds can be modeled by an exponential function. If A0 represents the original area of the wound and if A equals the area of the wound, then the function A n A e n 0 0.35 ( ) = − describes the area of a wound after n days following an injury when no infection is present to retard the healing. Suppose that a wound initially had an area of 100 square millimeters. (a) If healing is taking place, how large will the area of the wound be after 3 days? (b) How large will it be after 10 days? 113. Advanced-stage Pancreatic Cancer The percentage of patients P who have survived t years after initial diagnosis of advanced-stage pancreatic cancer is modeled by the function ( ) = ⋅ P t 100 0.3t Source: Cancer Treatment Centers of America (a) According to the model, what percent of patients survive 1 year after initial diagnosis? (b) What percent of patients survive 2 years after initial diagnosis? (c) Explain the meaning of the base 0.3 in the context of this problem. 114. Endangered Species In a protected environment, the population P of a certain endangered species recovers over time t (in years) according to the model ( ) = ⋅ P t 30 1.149x (a) What is the size of the initial population of the species? (b) According to the model, what will be the population of the species in 5 years? (c) According to the model, what will be the population of the species in 10 years? (d) According to the model, what will be the population of the species in 15 years? (e) What is happening to the population every 5 years? 115. Drug Medication The function ( ) = − D h e5 h 0.4 can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. How many milligrams will be present after 1 hour? After 6 hours?
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