SECTION 5.6 Logarithmic and Exponential Equations 343 (b) According to this model, when will the population of the United States be 470 million people? Source: U.S. Census Bureau 112. A Population Model The population of the world in 2024 was 8.05 billion people and was growing at a rate of 0.8%. per year. Assuming that this growth rate continues, the model ( ) ( ) = − P t 8.05 1.008 t 2024 represents the population P (in billions of people) in year t. (a) According to this model, when will the population of the world be 9 billion people? (b) According to this model, when will the population of the world be 12.5 billion people? Source: U.S. Census Bureau 113. Depreciation The value V of a Chevy Cruze LS that is t years old can be modeled by ( ) ( ) = V t 19,200 0.82 . t (a) According to the model, when will the car be worth $12,000? (b) According to the model, when will the car be worth $9000? (c) According to the model, when will the car be worth $3000? Source: Kelley Blue Book 114. Depreciation The value V of a Honda Civic LX that is t years old can be modeled by ( ) ( ) = V t 19,705 0.848 . t Credit: Grzegorz Czapski/Shutterstock (a) According to the model, when will the car be worth $14,000? (b) According to the model, when will the car be worth $10,000? (c) According to the model, when will the car be worth $7500? Source: Kelley Blue Book 115. Wine and Price Do more expensive wines taste better? In a study, both expert wine tasters and nonexperts were asked to rate a variety of wines on a scale from 1 to 4, with a higher number indicating a better tasting wine. For experts, the relation between rating, y, and price, x, was found to be = + y x 0.09ln 1.889. For nonexperts, the relation between rating, y, and price, x, was found to be = − + y x 0.048ln 2.337. (a) What is the rating of wine priced at $50 per bottle for the experts? (b) What is the rating of wine priced at $50 per bottle for the nonexperts? (c) What is the price of a wine rated 2.5 for the experts? (d) What is the price of a wine rated 2.5 for the nonexperts? 102. ( ) ( ) = + f x x log 5 3 and ( ) ( ) = − g x x log 1 . 3 (a) Solve ( )= f x 2. What point is on the graph of f ? (b) Solve ( )= g x 3. What point is on the graph of g? (c) Solve ( ) ( ) = f x g x . Do the graphs of f and g intersect? If so, where? (d) Solve ( )( ) + = f g x 3. (e) Solve ( )( ) − = f g x 2. 103. (a) If ( ) = + f x 3x 1 and ( ) = + g x 2 , x 2 graph f and g on the same Cartesian plane. (b) Find the point(s) of intersection of the graphs of f and g by solving ( ) ( ) = f x g x . Round answers to three decimal places. Label any intersection points on the graph drawn in part (a). (c) Based on the graph, solve ( ) ( ) > f x g x . 104. (a) If ( ) = − f x 5x 1 and ( ) = + g x 2 , x 1 graph f and g on the same Cartesian plane. (b) Find the point(s) of intersection of the graphs of f and g by solving ( ) ( ) = f x g x . Label any intersection points on the graph drawn in part (a). (c) Based on the graph, solve ( ) ( ) > f x g x . 105. (a) Graph ( ) = f x 3x and ( ) = g x 10 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, ( ) = f x 3 ,x and ( ) = g x 10 on the graph drawn in part (a). (c) Solve ( ) ( ) = f x g x and label the point of intersection on the graph drawn in part (a). 106. (a) Graph ( ) = f x 2x and ( ) = g x 12 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, ( ) = f x 2 ,x and ( ) = g x 12 on the graph drawn in part (a). (c) Solve ( ) ( ) = f x g x and label the point of intersection on the graph drawn in part (a). 107. (a) Graph ( ) = + f x 2x 1 and ( ) = − + g x 2 x 2 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, ( ) = + f x 2 , x 1 and ( ) = − + g x 2 x 2 on the graph drawn in part (a). (c) Solve ( ) ( ) = f x g x and label the point of intersection on the graph drawn in part (a). 108. (a) Graph ( ) = − + f x 3 x 1 and ( ) = − g x 3x 2 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, ( ) = − + f x 3 , x 1 and ( ) = − g x 3x 2 on the graph drawn in part (a). (c) Solve ( ) ( ) = f x g x and label the point of intersection on the graph drawn in part (a). 109. (a) Graph ( ) = − f x 2 4. x (b) Find the zero of f. (c) Based on the graph, solve ( ) < f x 0. 110. (a) Graph ( ) = − g x 3 9. x (b) Find the zero of g. (c) Based on the graph, solve ( ) > g x 0. 111. A Population Model The resident population of the United States in 2024 was 337 million people and was growing at a rate of 0.6% per year. Assuming that this growth rate continues, the model ( ) ( ) = − P t 337 1.006 t 2024 represents the population P (in millions of people) in year t. (a) According to this model, when will the population of the United States be 415 million people? (continued)
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