SECTION 5.4 Logarithmic Functions 325 (b) Determine how many minutes are needed for the probability to reach 80%. (c) Is it possible for the probability to equal 100%? Explain. 126. 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). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 5:00 pm. ( ) = − − F t e 1 t 0.15 (a) Determine how many minutes are needed for the probability to reach 50%. (b) Determine how many minutes are needed for the probability to reach 80%. 127. 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 was administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections? 128. Spreading of Rumors A model for the number N of people in a college community who have heard a certain rumor is ( ) ( ) = − − N d 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 days will elapse before 450 students have heard the rumor? 129. Current in an RL Circuit The equation governing the amount of current I (in amperes) after time t (in seconds) in a simple RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is [ ] = − ( ) − I E R e 1 R L t If = = E R 12 volts, 10 ohms, and = L 5 henrys, how long does it take to obtain a current of 0.5 ampere? Of 1.0 ampere? Graph the equation. 130. Learning Curve Psychologists sometimes use the function ( ) ( ) = − − L t A e 1 kt to measure the amount L learned at time t. Here A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes. (a) Determine the rate of learning k. (b) Approximately how many words will the student have learned after 10 minutes? (c) After 15 minutes? (d) How long does it take for the student to learn 180 words? The Richter Scale Problems 135 and 136 use the following discussion: The Richter scale is one way of converting seismographic readings into numbers that provide an easy reference for measuring the magnitude M of an earthquake. All earthquakes are compared to a zero-level earthquake whose seismographic reading measures 0.001 millimeter at a distance of 100 kilometers from the epicenter. An earthquake whose seismographic reading measures x millimeters has magnitude ( ) M x , given by ( ) = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ M x x x log 0 where = − x 10 0 3 is the reading of a zero-level earthquake the same distance from its epicenter. In Problems 135 and 136, determine the magnitude of each earthquake. 135. Magnitude of an Earthquake Mexico City in 1985: seismographic reading of 125,892 millimeters 100 kilometers from the center 136. Magnitude of an Earthquake San Francisco in 1906: seismographic reading of 50,119 millimeters 100 kilometers from the center 137. Alcohol and Driving The concentration of alcohol in a person’s bloodstream is measurable. Suppose that the relative risk R of having an accident while driving a car can be modeled by an equation of the form = R ekx where x is the percent concentration of alcohol in the bloodstream and k is a constant. (a) Suppose that a concentration of alcohol in the bloodstream of 0.03 percent results in a relative risk of an accident of 1.4. Find the constant k in the equation. Loudness of Sound Problems 131–134 use the following discussion:The loudness ( ) L x , measured in decibels ( ) dB , of a sound of intensity x, measured in watts per square meter, is defined as ( ) = L x x I 10 log , 0 where = − I 10 0 12 watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. 131. Normal conversation: intensity of = − x 10 7 watt per square meter. 132. Amplified rock music: intensity of − 10 1 watt per square meter. 133. Heavy city traffic: intensity of = − x 10 3 watt per square meter. 134. Diesel truck traveling 40 miles per hour 50 feet away: intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away, whose loudness is 70 decibels. (continued)
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