Exponential & Logarithmic Frank C. Wilson Functions I by file Activity Collection m Credit Card Balance Transfer DVD Player Sales Government Employee Salaries Living Longer Low Interest or Cash Back Shopping Center Planning The Bank Pays You The Bank Pays You #2 The Cost of Living The Cost of Living #2 Sa ple Featuring real-world contexts: www.makeitreallearning.com
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Table of Contents Introduction... 4 Activity Objectives... 5 Credit Card Balance Transfer: Working with Financial Formulas... 6 Solutions... 8 DVD Player Sales: Working with Logarithms... 10 Solutions... 12 Government Employee Salaries: Working with Percentage Change... 14 Solutions... 16 Living Longer: Using Exponential Function Models... 18 Solutions... 20 Low Interest or Cash Back: Using Financial Formulas... 22 Solutions... 24 Shopping Center Planning: Looking at Exponential and Linear Models... 26 Solutions... 28 The Bank Pays You: Using Compound Interest... 30 Solutions... 32 The Bank Pays You #2: Using Compound Interest... 34 Solutions... 36 The Cost of Living: Working with Exponential Models... 38 Solutions... 40 The Cost of Living #2: Working with Exponential Models... 42 Solutions... 44 About the Author... 46 Other Books in the Make It Real Learning Series... 46
Introduction When am I ever going to use this? It is a question that has plagued teachers and learners for decades. Now, with the help of the Make It Real Learning workbook series, you can answer the question. The Exponential and Logarithmic Functions I workbook focuses on real-world situations that may be effectively analyzed using exponential and logarithmic functions. From determining whether it is better to take a low interest rate or cash back to forecasting the cost of living, learners get to use mathematics in meaningful ways. Rest assured that each activity integrates real world information not just realistic data. These are real companies (e.g. Chase Bank, GMAC) and real world issues. The mathematical objectives of each activity are clearly specified on the Activity Objectives page following this introduction. Through the workbook series, we have consistently sought to address the content and process standards of the National Council of Teachers of Mathematics. There are multiple ways to use the activities in a teaching environment. Many teachers find that the activities are an excellent tool for stimulating mathematical discussions in a small group setting. Due to the challenging nature of each activity, group members are motivated to brainstorm problem solving strategies together. The interesting real world contexts motivate them to want to solve the problems. The activities may also be used for individual projects and class-wide discussions. As a ready-resource for teachers, the workbook also includes completely worked out solutions for each activity. To make it easier for teachers to assess student work, the solutions are included on a duplicate copy of each activity. We hope you enjoy the activities! We continue to increase the number of workbooks in the Make It Real Learning workbook series. Please visit www.makeitreallearning.com for the most current list of activities. Thanks! Frank C. Wilson Author 4
Exponential and Logarithmic Functions I Activity Objectives Activity Title Credit Card Balance Transfer: Working with Financial Formulas (p. 6) DVD Player Sales: Working with Logarithms (p. 10) Government Employee Salaries: Working with Percentage Change (p. 14) Living Longer: Using Exponential Function Models (p. 18) Low Interest or Cash Back: Using Financial Formulas (p. 22) Shopping Center Planning: Looking at Exponential and Linear Models (p. 26) The Bank Pays You: Using Compound Interest (p. 30) The Bank Pays You #2: Using Compound Interest (p. 34) The Cost of Living: Working with Exponential Models (p. 38) The Cost of Living #2: Working with Exponential Models (p. 42) Mathematical Objectives Use the present value of an annuity formula Solve exponential equations with logarithms Interpret graphs of functions Solve an exponential equation with logarithms Find the inverse of an exponential model Evaluate and graph a logarithmic function Calculate the average annual percentage change of a data set Determine average annual growth factors Solve exponential equations using logarithms Use regression to find an exponential model Interpret the meaning of the initial value and growth factor Solve an exponential equation with logarithms Calculate the monthly payment from the annuity formula Solve an exponential equation with logarithms Evaluate an exponential function at a given value Find the growth factor of an exponential function from a table Create an exponential model for a data set algebraically Use regression to find a linear model Interpret the meaning of the slope of a linear function Use the compound interest formula Solve a power function equation using rational exponents Find the annual percentage yield for a given interest rate Use the compound interest formula Solve an exponential function with logarithms Solve a power function equation by using rational exponents Find the point of intersection of two exponential equations Find the exponential growth factor from a verbal description Create an exponential model algebraically Evaluate an exponential equation at a given value Find the exponential growth factor from a verbal description Create an exponential model algebraically Solve an exponential equation using logarithms 5
Credit Card Balance Transfers Working with Financial Formulas I n June 2008, a consumer had an $11,894.80 balance on a Pentagon Federal Credit Union (PFCU) Visa card with a 0.499% monthly periodic rate (5.990% APR) as shown. A Chase Bank employee contacted the consumer and offered to transfer the balance to a Chase Bank credit card. The Chase card offered 0% interest for 15 months and a monthly periodic rate of 0.666% (7.99% APR) thereafter. The Chase card did not charge a balance transfer fee. 1. Assume that the consumer makes $250 monthly payments on the credit card and does not charge anything else to the card. a. Rounded to the nearest dollar, what will be the balance on the PFCU Visa card after 15 months? We use the present value of a decreasing annuity formula 1 ( 1+ i) PV = FV ( 1 + i) + PMT i 15 15 1 ( 1 + 0.00499) 11,894.80 = FV ( 1 + 0.00499) + 250 0.00499 11,894.80 FV ( 0.9281) + 3604.44 8290.36 FV ( 0.9281) FV $8933 After 15 months, the balance of the PFCU Visa will be about $8933. b. If he transfers his debt to the Chase card, what will be the balance on that card after 15 months? ( ) ( ) Balance in 15 months = Current Balance 15 monthly payments = $11,894.80 15 $250 = $8,144.80 6
3. After the first 15 months of payments have been made, the Chase credit card begins to charge interest. How long will it take to pay off the card if $250 payments continue to be made? From (2b) we know that the balance on the Chase card is $8144.80 after the first 15 payments have been made. 1 ( 1+ i) PV = PMT i 1 ( 1 + 0.00666) 8,144.80 = 250 0.00666 ( 0.00666)( 8144.80) = 250( 1 ( 1.00666) ) ( 0.00666)( 8144.80) = 1 ( 1.00666) 250 ( 0.00666)( 8144.80) 1 = ( 1.00666) 250 0.7830 ( 1.00666) 0.7830 ( 1.00666) ln ( 0.7830) ln ( 1.00666) ln ( 0.7830) ln( 1.00666) ln ( 0.7830) m = ln ( 1.00666) 36.8 37 It will take an additional 37 payments to pay off the card (52 payments in all). 4. The graph shows the balance on each card as the number of payments increases. Which of the cards will be paid off the quickest? Why? From the graph, we see that the Chase card is paid off in about 52 payments while the PFCU card is paid off in about 55 payments. Thus the Chase card is paid off quickest. Although the Chase card had a higher final interest rate (7.99% vs. 5.99%), the introductory rate of 0% for 15 months allowed us to rapidly reduce the amount owed. Consequently, the card was able to be paid off quickest even though it had the higher final interest rate. 7
Credit Card Balance Transfers Working with Financial Formulas I n June 2008, a consumer had an $11,894.80 balance on a Pentagon Federal Credit Union (PFCU) Visa card with a 0.499% monthly periodic rate (5.990% APR) as shown. A Chase Bank employee contacted the consumer and offered to transfer the balance to a Chase Bank credit card. The Chase card offered 0% interest for 15 months and a monthly periodic rate of 0.666% (7.99% APR) thereafter. The Chase card did not charge a balance transfer fee. 1. Assume that the consumer makes $250 monthly payments on the credit card and does not charge anything else to the card. a. Rounded to the nearest dollar, what will be the balance on the PFCU Visa card after 15 months? We use the present value of a decreasing annuity formula 1 ( 1+ i) PV = FV ( 1 + i) + PMT i 15 15 1 ( 1 + 0.00499) 11,894.80 = FV ( 1 + 0.00499) + 250 0.00499 11,894.80 FV ( 0.9281) + 3604.44 8290.36 FV ( 0.9281) FV $8933 After 15 months, the balance of the PFCU Visa will be about $8933. b. If he transfers his debt to the Chase card, what will be the balance on that card after 15 months? ( ) ( ) Balance in 15 months = Current Balance 15 monthly payments = $11,894.80 15 $250 = $8,144.80 8
3. After the first 15 months of payments have been made, the Chase credit card begins to charge interest. How long will it take to pay off the card if $250 payments continue to be made? From (2b) we know that the balance on the Chase card is $8144.80 after the first 15 payments have been made. 1 ( 1+ i) PV = PMT i 1 ( 1 + 0.00666) 8,144.80 = 250 0.00666 ( 0.00666)( 8144.80) = 250( 1 ( 1.00666) ) ( 0.00666)( 8144.80) = 1 ( 1.00666) 250 ( 0.00666)( 8144.80) 1 = ( 1.00666) 250 0.7830 ( 1.00666) 0.7830 ( 1.00666) ln ( 0.7830) ln ( 1.00666) ln ( 0.7830) ln( 1.00666) ln ( 0.7830) m = ln ( 1.00666) 36.8 37 It will take an additional 37 payments to pay off the card (52 payments in all). 4. The graph shows the balance on each card as the number of payments increases. Which of the cards will be paid off the quickest? Why? From the graph, we see that the Chase card is paid off in about 52 payments while the PFCU card is paid off in about 55 payments. Thus the Chase card is paid off quickest. Although the Chase card had a higher final interest rate (7.99% vs. 5.99%), the introductory rate of 0% for 15 months allowed us to rapidly reduce the amount owed. Consequently, the card was able to be paid off quickest even though it had the higher final interest rate. 9