324 CHAPTER 5 Exponential and Logarithmic Functions Applications and Extensions 121. Chemistry The pH of a chemical solution is given by the formula [ ] = − + pH log H 10 where [ ] +H is the concentration of hydrogen ions in moles per liter.Values of pH range from 0 (acidic) to 14 (alkaline). (a) What is the pH of a solution for which [ ] +H is 0.1? (b) What is the pH of a solution for which [ ] +H is 0.01? (c) What is the pH of a solution for which [ ] +H is 0.001? (d) What happens to pH as the hydrogen ion concentration decreases? (e) Determine the hydrogen ion concentration of an orange ( ) = pH 3.5 . (f) Determine the hydrogen ion concentration of human blood ( ) = pH 7.4 . 122. Diversity Index Shannon’s diversity index is a measure of the diversity of a population.The diversity index is given by the formula ( ) = − + + + H p p p p p p log log log n n 1 1 2 2 where p1 is the proportion of the population that is species 1, p2 is the proportion of the population that is species 2, and so on. In this problem, the population is people in the United States and the species is race. (a) According to the U.S. Census Bureau, the distribution of race in the United States in 2022 was: Race Proportion White 0.580 Black or African American 0.119 American Indian and Alaska Native 0.005 Asian 0.059 Native Hawaiian and Other Pacific Islander 0.002 Hispanic 0.192 Two or More Races 0.043 Source: U.S. Census Bureau Compute the diversity index of the United States in 2022. (b) The largest value of the diversity index is given by ( ) = H S log , max where S is the number of categories of race. Compute H . max (c) The evenness ratio is given by = E H H , H max where ≤ ≤ E 0 1. H If = E 1, H there is complete evenness. Compute the evenness ratio for the United States. (d) Obtain the distribution of race for the United States in 2010 from the Census Bureau. Compute Shannon’s diversity index. Is the United States becoming more diverse? Why? 123. Atmospheric Pressure The atmospheric pressure p on an object 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 height of an aircraft if the atmospheric pressure is 320 millimeters of mercury. (b) Find the height of a mountain if the atmospheric pressure is 667 millimeters of mercury. 124. 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, after how many days will the wound be one-half its original size? (b) How long before the wound is 10% of its original size? 125. 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 will arrive within t minutes of 12:00 pm. ( ) = − − F t e 1 t 0.1 (a) Determine how many minutes are needed for the probability to reach 50%. 103. = e 10 x3 104. = −e 1 3 x2 105. = + e 8 x2 5 106. = − + e 13 x2 1 107. ( ) + = x log 4 2 7 2 108. ( ) + + = x x log 4 2 5 2 109. = − log 8 6 x 2 110. = − log 3 1 x 3 111. = e5 7 x 0.2 112. ⋅ = − 8 10 3 x2 7 113. ⋅ = − 2 10 5 x 2 114. = + e4 5 x 1 115. Suppose that ( ) ( ) = + − G x x log 2 1 2. 3 (a) What is the domain of G? (b) What is G( ) 40 ? What point is on the graph of G? (c) If ( ) = G x 3, what is x? What point is on the graph of G? (d) What is the zero of G? 116. Suppose that ( ) ( ) = + − F x x log 1 3. 2 (a) What is the domain of F? (b) What is ( ) F 7 ? What point is on the graph of F? (c) If ( ) = − F x 1, what is x? What point is on the graph of F? (d) What is the zero of F? Mixed Practice In Problems 117–120, graph each function. Based on the graph, state the domain and the range, and find any intercepts. 117. ( ) ( ) = − < > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x ln if 0 ln if 0 118. ( ) ( ) ( ) = − ≤− − − − < < ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x ln if 1 ln if 1 0 119. ( ) = − < < ≥ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x ln if 0 1 ln if 1 120. ( ) = < < − ≥ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x ln if 0 1 ln if 1
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