People, Tea, and Carbon Dioxide

Transcription

1 Lesson.1 Skills Practice Name Date People, Tea, and Carbon Dioide Modeling Using Eponential Functions Problem Set For each given data set determine the eponential regression equation and the value of the correlation coefficient, r. Round all values to the hundredths place f() (1.06 ) r Chapter Skills Practice 747

2 Lesson.1 Skills Practice page Evaluate each function for the given value of. Round our answer to the hundredths place. 7. Evaluate f() (1.05 ) when 5 6. f(6) (1.05 ) Evaluate f() 5 55(0.82 ) when Evaluate f() 5 200(1.11 ) when Evaluate f() 5 10(2 ) when Evaluate f() (0.99 ) when Evaluate f() 5 0.5(1.094 ) when Evaluate f() (0.485 ) when Evaluate f() 5 180(0.35 ) when Evaluate f() 5 2.5(1.5 ) when Evaluate f() (0.95 ) when Chapter Skills Practice

3 Lesson.1 Skills Practice page 3 Name Date Determine the eponential regression equation that models each situation. Use the equation to make the associated prediction. Round all values to the hundredths place. 17. Tamara deposited $500 into a savings account in The table shows the value of Tamara s savings account from 1970 to Predict the account s value in Time Since 1970 (ears) Account Value (dollars) f() (1.06 ) f(50) (1.06 ) The account s value will be approimatel $96.42 in Tamika deposited $1000 into a savings account in The table shows the value of Tamika s savings account from 1980 to Predict when the account s value will be $5000. Time Since 1980 (ears) Account Value (dollars) A marine biologist monitors the population of sunfish in a small lake. He records 800 sunfish in his first ear, 600 sunfish in his fourth ear, 450 sunfish in his sith ear, and 350 sunfish in his tenth ear. Predict the population of sunfish in the lake in his siteenth ear. Chapter Skills Practice 749

4 Lesson.1 Skills Practice page A marine biologist monitors the population of catfish in a small lake. He records 50 catfish in his first ear, 170 catfish in his fourth ear, 380 catfish in his sith ear, and 1900 catfish in his tenth ear. Predict when the population of catfish in the lake will be Ever hour, a scientist records the number of cells in a colon of bacteria growing in her lab. The sample begins with 15 cells. Predict the number of cells in the colon after 7 hours. Hour Number of Cells Ever hour, a scientist records the number of cells in a colon of bacteria growing in her lab. The sample begins with 50 cells. Predict how long it will take the sample to grow to 2000 cells. Hour Number of Cells Chapter Skills Practice

5 Lesson.2 Skills Practice Name Date Stop! What Is Your Reaction? Modeling Stopping Distances and Reaction Times Problem Set Determine the regression equation which best fits each given data set. Create a scatter plot of the data and graph the regression equation on the same grid. 1. The table shows the height of a suppl kit dropped from a helicopter at various times. Time Since Release (seconds) Height (feet) The quadratic regression equation f() , where the function represents the height of the suppl kit seconds after it is released, best fits the data Height (feet) Time Since Release (seconds) Chapter Skills Practice 751

6 Lesson.2 Skills Practice page 2 2. The table shows the population of river otters in a river over a period of time. Time Since 2005 (ears) River Otter Population Chapter Skills Practice

7 Lesson.2 Skills Practice page 3 Name Date 3. The table shows a runner s speed at different distance markers during a marathon. Distance (miles) Runner s Speed (miles per hour) Chapter Skills Practice 753

8 Lesson.2 Skills Practice page 4 4. The table shows the population of bald eagles in a particular count over a period of time. Time Since 2000 (ears) Bald Eagle Population Chapter Skills Practice

9 Lesson.2 Skills Practice page 5 Name Date 5. The table shows the population of ash trees in a forest over a period of time. Time Since 1980 (ears) Ash Tree Population Chapter Skills Practice 755

10 Lesson.2 Skills Practice page 6 6. The table shows the height of a burning candle over a period of time. Time (minutes) Height (centimeters) Chapter Skills Practice

11 Lesson.2 Skills Practice page 7 Name Date The function modeling each problem situation is given. Sketch the inverse of each function on the grid provided and label the aes. Then, describe the domain and range of both functions as the relate to the problem situation. 7. The function models a mountain climber s elevation over his 10-da mountain climb. Elevation (feet) 18,000,000 14,000 12,000 10, Time Since Starting Climb (das) Time Since Starting Climb (das) Elevation (thousands of feet) For the given function, the domain is all real numbers greater than or equal to 0 and less than or equal to 10, because it represents the time, in das, since the climber started his climb. The range is all real numbers greater than or equal to 0 and less than or equal to about 15,800, because it represents the climber s elevation from his starting point of 0 feet to his ending point of about 15,800 feet. For the inverse function, the domain is all real numbers greater than or equal to 0 and less than or equal to about 15,800, because it represents the climber s elevation from his starting point of 0 feet to his ending point of about 15,800 feet. The range is all real numbers greater than or equal to 0 and less than or equal to 10, because it represents the time, in das, since the climber started his climb. Chapter Skills Practice 757

12 Lesson.2 Skills Practice page 8 8. In the ear 1990, park rangers stocked an unpopulated lake with 50 channel catfish. The function models the channel catfish population over a period of time. Channel Catfish Population Time Since 1990 (ears) 758 Chapter Skills Practice

13 Lesson.2 Skills Practice page 9 Name Date 9. Twent ears ago, Daniel deposited $ in a savings account. The function models the balance in the savings account in dollars Balance (dollars) Time (ears) Chapter Skills Practice 759

14 Lesson.2 Skills Practice page On a da when he does not eercise, Carl drinks 24.3 ounces of water. The function models the amount of water Carl drinks while eercising. Amount of Water (ounces) Time Spent Eercising (hours) 760 Chapter Skills Practice

15 Lesson.2 Skills Practice page 11 Name Date 11. The function models the relationship between the number of wool hats sold and the amount of mone earned. 18 Amount Earned (dollars) Number of Hats Sold Chapter Skills Practice 761

16 Lesson.2 Skills Practice page The function models a basketball plaer s free throw percentage during the course of the season. 90 Free Throw Percentage (%) Number of Games Plaed 762 Chapter Skills Practice

17 Lesson.2 Skills Practice page 13 Name Date The graph displas the average scores of bowlers in the Spare Time Bowling League throughout their 10-game season. The function L() models the left-handed bowlers average scores after games. The function R() models the right-handed bowlers average scores after games. Analze the graph to answer each question. Average Bowling Scores Number of Games L() R() 13. Which function famil does each function appear to belong to? The function L() appears to be a quadratic function, because it is a smooth curve which decreases and then increases over the domain. The function R() appears to be a linear function, because it is a straight line. 14. Describe how each group s average scores changed throughout the season. Chapter Skills Practice 763

18 Lesson.2 Skills Practice page At which point(s) during the season did both groups have the same average scores?. Consider the function A(), where A() 5 L() 2 R(). What does A() mean in terms of the problem situation? What does it mean for the value of A() to be equal to 0? 764 Chapter Skills Practice

19 Lesson.3 Skills Practice Name Date Modeling Data Helps Us Make Predictions Using Quadratic Functions to Model Data Problem Set A quadratic regression equation is given for the amount of profit, P, in dollars that a number of produced items, n, generates. Predict the profit for each given number of produced items. 1. P 5 2n n for n 5 20 P 5 2(20) (20) P P The profit when 20 items are produced should be $ P 5 2n n for n P 5 22n n for n P 5 22n n for n Chapter Skills Practice 765

20 Lesson.3 Skills Practice page 2 5. P 5 24n n for n P 5 24n n for n Use a graphing calculator to determine the quadratic regression equation for each data set. Round decimals to the nearest thousandth Chapter Skills Practice

21 Lesson.3 Skills Practice page 3 Name Date Determine the -intercept, -intercept(s), and verte of the graph of each given quadratic regression equation. Then determine what these values mean in terms of the problem situation and tell whether the values make sense. 13. The speed, s, of a car and the car s average fuel efficienc, G, in miles per gallon at that speed can be modeled b the linear regression equation G s s The -intercept is the point (0, ). The -intercept represents the fuel efficienc of the car when it is traveling at 0 miles per hour (at a stop). Because the fuel efficienc is a negative number, this value does not make sense. The -intercepts are the points (6.707, 0) and ( , 0). The -intercepts represent the speed of the car when the fuel efficienc is zero. The fuel efficienc would be zero when the car is not moving, not when the car is traveling miles per hour or miles per hour. So, these values do not make sense. The verte is the point (53.643, ). The verte represents the speed at which the car is getting the highest fuel efficienc. So, when the car is traveling at miles per hour, it gets its highest fuel efficienc, which is miles per gallon. This value makes sense. Chapter Skills Practice 767

22 Lesson.3 Skills Practice page An athlete throws a disc upward at an angle. The height in feet, h, of the disc can be modeled b the linear regression equation h , where represents the distance in feet that the disc has traveled horizontall. 15. Martin and his friend are plaing catch with a baseball. Martin tosses the ball to his friend, but he overthrows it, and it hits the ground. The height in feet, h, of the baseball can be modeled b the quadratic regression equation h , where represents the distance in feet that the baseball has traveled horizontall. 768 Chapter Skills Practice

23 Lesson.3 Skills Practice page 5 Name Date. A compan s profit in dollars, p, can be modeled b the quadratic regression equation p , , where represents the number of ears since the compan was started. 17. The temperatures, t, in degrees Fahrenheit recorded during a 10-hour winter snow storm can be modeled b the quadratic regression equation t where represents the number of hours the storm has lasted. 18. A compan s dail profit, p, from selling calculators can be modeled b the quadratic regression equation p , where represents the price of the calculator. Chapter Skills Practice 769

24 Lesson.3 Skills Practice page 6 Use the given regression equation to answer each question. 19. A farmer finds that his crop ield per acre can be modeled b the quadratic regression equation , where represents the amount of fertilizer applied in pounds per hundred square feet and represents the crop ield in bushels. What is the approimate ield when 20 pounds of fertilizer are applied per hundred square feet? (20) (20) The farmer should have a ield of bushels per hundred square feet. 20. The growth of sobean plants in inches over a certain time is tested with different amounts of fertilizer. The growth can be modeled b the quadratic regression equation , where represents the growth in inches and represents the amount of fertilizer per plant in milligrams. How much growth could be epected from a plant given 40 milligrams of fertilizer? 21. The height of a ball thrown at an angle is measured photographicall. The height of the ball can be modeled b the quadratic regression equation h 5 2.2t t 1 4, where h is the height of the ball after t seconds. At what time should the ball reach a height of 30 feet? 22. The quadratic regression equation models the relationship between the selling price of a necklace,, in dollars and the profit earned each month,, in dollars. To reach a profit of $300 per month from the necklaces, what should the selling price be? 770 Chapter Skills Practice

25 Lesson.3 Skills Practice page 7 Name Date 23. A compan manufactures biccles. The relationship between the number of biccles made and the cost to produce those biccles can be modeled b the quadratic regression equation , where represents the total cost to make the biccles and represents the number of biccles made. What would be the cost to manufacture 50 biccles? 24. A model rocket is launched into the air. The height of the model rocket can be modeled b the quadratic regression equation , where represents the height of the model rocket in feet and represents the time in seconds. At what time should the model rocket reach a height of 2000 feet? Chapter Skills Practice 771

26 772 Chapter Skills Practice

27 Lesson.4 Skills Practice Name Date BAC Is BAD News Choosing a Function to Model BAC Problem Set Use a graphing calculator to determine the indicated regression equation for each data set. Eplain the meaning of each variable in the regression equation. 1. Determine a quadratic regression equation that shows the relationship between the length of time in months that a new television has been for sale and the price of the television in dollars. Time (months) Price (dollars) represents the time in months the television has been for sale represents the price in dollars of the television 2. Determine a quadratic regression equation that shows the relationship between the number of ears since a collector purchased an antique table and the value of the table in dollars. Time (ears) Price (dollars) , Determine an eponential regression equation that shows the relationship between the time in months and the population of a town. Time (months) Population Chapter Skills Practice 773

28 Lesson.4 Skills Practice page 2 4. Determine an eponential regression equation that shows the relationship between the time in months and the rabbit population in a park. Time (months) Rabbit Population Determine a quadratic regression equation that shows the relationship between a vehicle s mileage in thousands of miles and the cost for repairs in dollars during the vehicle s last inspection. Mileage (thousands of miles) Cost for Repairs (dollars) Determine an eponential regression equation that shows the relationship between the time in ears and the amount of interest on a mortgage in dollars. Time (ears) Interest (dollars) , Chapter Skills Practice

29 Lesson.4 Skills Practice page 3 Name Date Determine which regression equation is the best fit for each set of data points. Eplain our reasoning. 7. Quadratic graph has equation Eponential graph has equation 5 (1.0851)(1.045 ) Population of Alban (thousands) Time (ears) The eponential regression equation fits the data better than the quadratic regression equation. The eponential regression equation is closer to more points in the data set. Chapter Skills Practice 775

30 Lesson.4 Skills Practice page 4 8. Quadratic graph has equation Eponential graph has equation 5 ( )( ) Temperature of a Pan ( F) Time (min) 9. Quadratic graph has equation Eponential graph has equation 5 (23.202)(1 ) Cost of Production (thousands of dollars) Number of DVDs (thousands) 776 Chapter Skills Practice

31 Lesson.4 Skills Practice page 5 Name Date 10. Quadratic graph has equation Eponential graph has equation 5 ( )( ) Amount of Radioactive Material (kg) Time (ears) Chapter Skills Practice 777

32 Lesson.4 Skills Practice page The quadratic regression equation is The eponential regression equation is 5 (133.56)(1.01). Total Traveling Distance (miles) Milepost Location 12. The quadratic regression equation is The eponential regression equation is ( ) Price (cents) Time (decades since 1900) 778 Chapter Skills Practice

33 Lesson.4 Skills Practice page 7 Name Date Use the given regression equation to answer each question. 13. The height of a ball in feet t seconds after it is projected upward can be modeled b the regression equation h(t) 5 2 t 2 1 0t 1 5. What is the height of the ball after 2 seconds? h(2) 5 2(2 ) 2 1 0(2) (4) The height of the ball is 261 feet after 2 seconds. 14. The population of a town after t ears can be modeled b the regression equation p(t) 5 15,000(1.07) t. What will the population of the town be after 5 ears? 15. A boiling pot of water is removed from a burner. Its temperature in degrees Fahrenheit can be modeled b the regression equation t() (0.98 ), where represents the number of minutes after the pot is removed from the burner. When does the temperature reach 152 degrees?. The balance in a bank account can be modeled b the regression equation b(t) 5 2 t t 1 400, where t represents the time in months. When is the balance in the bank account $382? 17. The population of polar bears in a park after t ears can be modeled b the regression equation p(t) 5 350(0.98 ) t. What is the population of the polar bears after 20 ears? Chapter Skills Practice 779

34 Lesson.4 Skills Practice page The height of a rocket in feet t seconds after it is launched can be modeled b the regression equation h(t) 5 2 t t What is the height of the rocket 40 seconds after it is launched? Ralph purchased a new 1970 Chev Nova in the ear it was manufactured for $3000. The scatter plot shows the car s value over a period of time since The quadratic regression equation that best fits the data is f() , where f() represents the value of the car in dollars and represents the time since 1970 in ears. The function is graphed on the grid. Analze this information to answer each question. Value of the Car (dollars) 18,000,000 14,000 12,000 10, Time Since 1970 (ears) 19. Discuss the domain and range of the function as the relate to the problem situation. The domain is all real numbers greater than or equal to 0, because the car did not have a value prior to the ear of its manufacture. The range is all real numbers greater than or equal to approimatel $1000, because the function s value starts at approimatel $3685, and then drops to approimatel $1000 before rising from that point onward. 20. Discuss the intervals of increase and decrease as the relate to the problem situation. 21. Discuss the - and -intercepts of the function as the relate to the problem situation. 780 Chapter Skills Practice

35 Lesson.4 Skills Practice page 9 Name Date 22. Discuss an minimums and maimums as the relate to the problem situation. 23. Predict the value of the car in Wh do ou think the value of this car is best represented b a quadratic function? Do ou think this is true of all cars? The scatter plot shows the number of wild Burmese pthons captured in a Florida count over a period of time. The eponential regression equation that best fits the data is p() (1.68 ), where p represents the number of wild Burmese pthons captured and represents the number of ears since The function is graphed on the grid. Analze this information to answer each question. Number of Pthons Captured Time Since 2005 (ears) 25. Discuss an minimums and maimums as the relate to the problem situation. The function does not have a maimum value. Even though the given eponential function has no minimum value, the function as it relates to the problem has a minimum value of approimatel 5 in the ear Chapter Skills Practice 781

36 Lesson.4 Skills Practice page Discuss the domain and range of the function as the relate to the problem situation. 27. Discuss the intervals of increase and decrease as the relate to the problem situation. 28. Discuss the - and -intercepts of the function as the relate to the problem situation. 29. Predict the number of wild Burmese pthons captured in Florida in Wh might the number of wild Burmese pthons captured b increasing eponentiall? 782 Chapter Skills Practice

37 Lesson.5 Skills Practice Name Date Cell Phone Batteries, Gas Prices, and Single Famil Homes Modeling with Piecewise Functions Problem Set Write a possible scenario to model each piecewise function graph. Determine the slope, -intercept(s) and -intercept and eplain what each means in terms of the problem situation. 1. The graph shows the depth at which a shark swims after a tracking device is attached to its dorsal fin. Answers will var. The shark is released into the water after having 9 the device attached. It swims downward to a 8 depth of 5 meters over a two minute period. It remains at this depth for 3 minutes before 7 D swimming upward to a depth of 2 meters over the 6 net 3 minutes. Finall, it swims downward past B 5 the a depth of 10 meters over the net 2 minutes. The slope in this situation represents the rate at 4 C which the shark s depth is changing over time. 3 A When the slope is positive, the shark is going 2 1 deeper. When the slope is negative, the shark is going towards the surface. When the slope is 0, the shark s depth is remaining constant. The slope of part A is 5, the slope of part B is 0, the 2 Time (minutes) slope of part C is 21, and the slope of part D is 4. The -intercept in this situation represents the time at which the shark is at the surface. The onl -intercept is (0, 0) which means the shark was onl at the surface at the moment it was released. The -intercept is (0, 0) and represents the depth of the shark when it was first released. In this case, the depth of the shark was 0 meters when it was released. Depth (meters) Chapter Skills Practice 783

38 Lesson.5 Skills Practice page 2 2. Mercedes leaves her beach house for a walk along the beach. The graph shows her distance from home for the duration of her walk. Distance from Home (miles) A B C D Time of Walk (hours) 3. The graph shows a hiker s elevation during his 10-hour mountain hike Elevation (feet) A D B C Time of Hike (hours) 784 Chapter Skills Practice

39 Lesson.5 Skills Practice page 3 Name Date 4. John goes for a bike ride earl in the morning and returns home later that da. The graph shows his distance from home for the duration of his ride. Distance (miles) C 5 B 4 3 A D Time (hours) Chapter Skills Practice 785

40 Lesson.5 Skills Practice page 4 5. Peton takes her dog for a long walk on Saturda. The graph shows her distance from home for the duration of the walk. Distance (miles) 9 8 C 7 6 B 5 D 4 3 A Time (hours) 6. Tona travels to her friend Aleandra s house, stas awhile, and then returns home. The graph shows her distance from home for the duration of her trip. C 4 B Distance (miles) A D Time (hours) 786 Chapter Skills Practice

41 Lesson.5 Skills Practice page 5 Name Date Determine a piecewise function to model each given data set and define the variables used. Then, graph the piecewise function on the grid provided. Round decimals to the nearest thousandth. 7. The table and the scatter plot show the number of emploees emploed b a compan over a period of time. Time Since 1995 (ears) Number of Emploees Number of Emploees Time Since 1995 (ears) Answers will var. Let f() represent the number of emploees emploed b the compan ears since f() , 0 # # , 4, # 9 Chapter Skills Practice 787

42 Lesson.5 Skills Practice page 6 8. The table and the scatter plot show the outdoor temperature in Washington, DC over a period of time. Time Since 8 am (hours) Temperature (8F) Temperature ( F) Time Since 8 AM (hours) 788 Chapter Skills Practice

43 Lesson.5 Skills Practice page 7 Name Date 9. The table and the scatter plot show the population of black bears in a Tennessee count over a period of time. Time Since 1980 (ears) Black Bear Population Black Bear Population Time Since 1980 (ears) Chapter Skills Practice 789

44 Lesson.5 Skills Practice page The table and the scatter plot show the height of a bouncing ball dropped from a building over a period of time. Time Since Release (seconds) Height of Ball (feet) Height of Ball (feet) Time Since Release (seconds) Chapter Skills Practice

45 Lesson.5 Skills Practice page 9 Name Date 11. The table and the scatter plot show a bookstore s earl profit over a period of time. Time Since 2001 (ears) Profit (dollars) 0 10, , , , , , , , , , ,000 Profit (dollars) 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10, Time Since 2001 (ears) Chapter Skills Practice 791

46 Lesson.5 Skills Practice page The table and the scatter plot show the number of earl subscriptions to a small town newspaper over a period of time. Time Since 1990 (ears) Number of Subscriptions Number of Subscriptions Time Since 1990 (ears) Chapter Skills Practice