College Algebra Lecture Notes Exponential Functions

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1 In 1988, a judge in Yonkers, New York instituted an exponential fine on the city of Yonkers Below is the background and scenario, published in the New York Times 1 : Dec 1, 1980: Justice Department sues Board of Education, City of Yonkers and Yonkers Community Development Agency, charging that the city racially discriminated in education and public housing Nov 20, 198: Judge Leonard B Sand of Federal District Court in Manhattan rules that Yonkers s housing and schools were intentionally segregated by race A housing remedy order directs the city to build 200 units of public housing and to plan additional subsidized housing Jan 28, 1988: City Council approves consent decree that sets timetable for building 200 units of public housing and commits city to an additional 800 subsidized units July 26, 1988: Court sets Aug 1 deadline for Council to adopt zoning amendment needed to build the 800 units Aug 1, 1988: Council rejects amendment in a 4-to-3 vote Aug 2, 1988: Judge Sand finds city and the four Councilmen who voted against the amendment in contempt of court and imposes fines The city s fines start at $100 and double every day The Councilmen s fines start at $00 a day and increase by $00 each day Group Work 1 Let P be the amount fined (in dollars) on day t Complete the first three entries of each Table 1 and Table 2 Table 1 Councilmen Table 2 City of Yonkers t P Formula t P Formula t t 1 1

2 Example 1 Figure 1 Councilmen Figure 2 City of Yonkers P, fine in dollars 3,00 3,000 2,00 2,000 1,00 1, t, time in days P, fine in dollars 3,600 3,200 2,800 2,400 2,000 1,600 1, t, time in days Group Work 2 On what day will will the city of Yonkers fine reach over $1,000,000? Group Work 3 How much will the city of Yonkers be fined on day 30? Councilmen s fines be on that day? What will each of the Instructor: AECary Page 2 of 8

3 Definition 1 An exponential function is of the form where a is the initial value b is the growth factor and b > 0 f(t) = a b t Consequently, an exponential function is a function that increases or decreases at a constant percent rate Let s review percent increase and decrease as we work through these examples Example 2 A compost pile has 27 cubic feet of waste and decays at a rate of 10% per month How much waste is left after 1 month? How much is left after 3 months? After t months? Let Q be the volume of compost in cubic feet at time t, in months since decay began Write the formula modeling this exponential decay: Instructor: AECary Page 3 of 8

4 Group Work 4 Complete Table 3 above Then graph those ordered pairs in Figure 3 Table 3 Exponential Compost Decay t Q Formula 30 Figure (09) = (27(09)) (09) = (27(09)(09)) (09) = Q, volume of compost in ft t, time in months 2 30 Group Work Does the decreasing exponential function in the last example ever reach zero? Why or why not? Example 3 Suppose the size of a poplulation of honeybees is initially P 0 Let P be the size of the population after t months If the population increases by 1% each month, how many bees will there be after 1 month? After 2 months? After 3 months? After t months? In this example, the factor 11 is known as the grows each month is known as the The percent by which the population Instructor: AECary Page 4 of 8

5 Example 4 You start a new job with an initial salary of $36,000 per year Each year thereafter, you receive a 3% raise Let S be your salary t years after you start your new job (a) Write S as a function of t Identify the growth factor and growth rate (b) What will your annual salary be after 1 year? (c) What will your salary be after years? (d) What will your salary be after 10 years? (e) When will your salary reach $0,000? (Use your graphing calculator to solve this) Instructor: AECary Page of 8

6 Example After caffeine is consumed, it leaves the body at a rate of about 16% per hour Assume 200 mg of caffeine are consumed (a) Write the milligrams of caffeine in the body Q, as a function of hours, t, since the caffeine was consumed Identify the growth factor and growth rate (b) How much caffeine will still be in the body 1 hour later? (c) How much caffeine will still be in the body 8 hours later? (d) How much caffeine will still be in the body 12 hours later? (e) When will there be less than 10 mg of caffeine remaining in the bloodstream? (Use your graphing calculator to solve this) Instructor: AECary Page 6 of 8

7 Group Work 6 In 2000, the population of Oregon was 342 million people Each year since then the population increased by an average of 1% 2 Let t be the number of years after 2000 and let P be the population of Oregon in millions (a) Assuming growth continues in this way, find the equation of an exponential function modeling this data Identify the growth factor and growth rate (b) Assuming this exponential growth continues, what will the population be in 2020? (c) Assuming this exponential growth continues, when will the population reach million people? (Use your graphing calculator to solve this) 2 Instructor: AECary Page 7 of 8

8 Group Work 7 The area of a square of paper is initially 100 square inches Each time the paper is folded in half, the area becomes half of the original area Let A be the area of the paper (in square inches) after it is folded t times (a) Write a formula representing the area A as a function of the number of times the paper is folded t Identify the growth factor and growth rate (b) Complete Table 4 and Figure 4 below t 0 1 A Table Figure A, area in sq in t, number of folds (c) From Table 4, what fold resulted in an area of less than one square inch? Instructor: AECary Page 8 of 8

Writing Exponential Equations Day 2

Writing Exponential Equations Day 2 MGSE9 12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational,