Chapter Test 379 60. Spreading of a Disease Jack and Diane live in a small town of 50 people. Unfortunately, both Jack and Diane have a cold. Those who come in contact with someone who has this cold will themselves catch the cold.The data that follow represent the number of people in the small town who have caught the cold after t days. Number of People with Cold, C Days, t 2 4 8 14 22 30 37 42 44 0 1 2 3 4 5 6 7 8 (a) Using a graphing utility, draw a scatter plot of the data. Comment on the type of relation that appears to exist between the number of days that have passed and the number of people with a cold. (b) Using a graphing utility, build a logistic model from the data. (c) Graph the function found in part (b) on the scatter plot. (d) According to the function found in part (b), what is the maximum number of people who will catch the cold? In reality, what is the maximum number of people who could catch the cold? (e) Sometime between the second and third day, 10 people in the town had a cold. According to the model found in part (b), when did 10 people have a cold? (f) How long will it take for 46 people to catch the cold? (b) Using a graphing utility, build an exponential model from the data. (c) Express the function found in part (b) in the form ( ) = N t N e . kt 0 (d) Graph the exponential function found in part (b) or (c) on the scatter plot. (e) Use the exponential function from part (b) or (c) to predict the population at = x 7 hours. (f) Use the exponential function from part (b) or (c) to predict when the population will reach 0.75. 59. Wind Chill Factor The data represent the wind speed (mph) and the wind chill factor at an air temperature of ° 15 F. Wind Chill Factor (ºF) Wind Speed (mph) 7 3 0 22 24 25 27 5 10 15 20 25 30 35 Source: U.S. National Weather Service (a) Using a graphing utility, draw a scatter plot with wind speed as the independent variable. (b) Using a graphing utility, build a logarithmic model from the data. (c) Using a graphing utility, draw the logarithmic function found in part (b) on the scatter plot. (d) Use the function found in part (b) to predict the wind chill factor if the air temperature is ° 15 F and the wind speed is 23 mph. The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. 1. If ( ) = + − f x x x 2 2 and ( ) = + g x x2 5, find: (a) f g and state its domain (b) ( ) − g f ( ) 2 (c) ( )( ) − f g 2 2. Determine whether the function is one-to-one. (a) = + y x4 3 2 (b) = + − y x 3 5 3. Find the inverse of ( ) = − f x x 2 3 5 and check your answer. State the domain and the range of f and −f .1 4. If the point ( ) − 3, 5 is on the graph of a one-to-one function f, what point must be on the graph of −f ?1 In Problems 5–7, solve each equation. 5. = 3 243 x 6. = log 16 2 b 7. = x log 4 5 In Problems 8–11, evaluate each expression without using a calculator. 8. log 1 36 6 9. log 10,000 10. 8log 52 11. e ln 7 In Problems 12 and 13, for each function f: (a) Find the domain of f. (b) Graph f. (c) From the graph of f, find the range and any asymptotes. (d) Find −f ,1 the inverse of f. (e) Find the domain and the range of −f .1 (f) Graph −f .1 12. ( ) = − + f x 4 2 x 1 13. ( ) ( ) = − − f x x 1 log 2 5 Chapter Test
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