3/6/2012 Math for Liberal Arts MAT 110: Chapter 8 Notes Growth: Linear vs. Exponential David J. Gisch February 28, 2012 Growth: Linear versus Exponential Growth: Linear versus Exponential • Linear Growth occurs when a quantity grows by some • Straightown grows by the same absolute amount each fixed absolute amount in each unit of time. year and Powertown grows by the same relative amount each year. • Exponential Growth occurs when a quantity grows by the same fixed relative amount—that is, by the same percentage—in each unit of time. 1
3/6/2012 Key Facts about Exponential Growth Growth: Linear versus Exponential • Exponential growth leads to repeated doublings. Example 8.A.1: Recall simple interest versus compound With each doubling, the amount of increase is interest. Simple interest is the same amount of interest approximately equal to the sum of all preceding every time as where compound interest is the same doublings. percent of interest at each step of time. For example, let's say we invested $500 with an interest rate of 10%. • Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, Year Sim ple Com pound exponentially growing quantities reach impossible 1 $1,000 $1,000 proportions. 2 $1,000+100=$1,100 $1,000+100=$1,100 3 $1,100+100=$1,200 $1,100+110=$1,210 4 $1,200+100=$1,300 $1,210+121=$1,331 5 $1,300+100=$1,400 $1,331+133.10=$1,464.10 Linear: We add the Exponential: We same amount, $100, add the same percent, every time. 10%, every time. Growth: Linear versus Exponential Growth: Linear versus Exponential Example 8.A.2: Bacteria in a Bottle: Suppose you put a Example 8.A.3: You are given a choice, take $1000 each single bacterium in a bottle at 11:00 a.m. It grows and at month for the rest of your life or be given a magic penny. 11:01, it divides into two bacteria. These two bacteria grow The magic penny will turn into two pennies after one day. and at 11:02 divide into four bacteria, which grow and at Then double again into four pennies the next day, and so 11:03 divide into eight bacteria, and so on. Thus, the on. Which option would you rather take? bacteria doubles every minute. After 30 Years! If the bottle is half-full at 11:59, when will the bottle be completely full? • $1,000 option: You have $1000 12 30 � $360,000 • Penny: You have $0.01�2� �� � $10,737,418.24 2
3/6/2012 Linear or Exponential? Linear or Exponential? Example 8.A.4: The price of milk is increasing by 3 cents Example 8.A.5: The price of a house is increasing by 2% per week. per year. (a) Is this exponential or linear? (a) Is this exponential or linear? (b) If the price of milk is $3.65 today, what will it be in 5 (b) If the price of the house is $175,000 today, what will it weeks? be in 5 years? Exponential Growth & Decay • The time required for each doubling in exponential growth is called doubling tim e . • The time required for each halving in exponential decay is called halving tim e . Doubling Time and Half-Life 3
3/6/2012 Doubling Time Doubling Time (Exponential Growth) • After a time � , an exponentially growing quantity with a Example 8.B.1: Recall this chart from the last section. Can doubling time of � ������ increases in size by a factor of you use the chart to create the formula for Powertown? 2 � � ������ ⁄ . The new value of the growing quantity is related to its initial value (at � � 0 ) by ��� ����� � ������� ����� � 2 � � ������ ⁄ Whatever unit of time is used to measure your doubling period is the unit of time you should use for � . For example, is a bacteria doubles every 8 hours then � must be measured in hours. Doubling Time (Exponential Growth) Doubling Time (Exponential Growth) Example 8.B.2: Using your equation, what will the Example 8.B.3: World Population Growth: World population be in 30 years? Does this match the chart? population doubled from 3 billion in 1960 to 6 billion in 2000. Suppose that the world population continued to ��� ����� � 10,000 � 2 � �� ⁄ grow (from 2000 on) with a doubling time of 40 years. What would be the population in 2050? ��� ����� � 10,000 � 2 �� �� ⁄ • Always identify your initial value and year first! � 441,636 4
3/6/2012 Doubling Time (Exponential Growth) Approximate Double Time Formula (The Rule of 70) For a quantity growing exponentially at a rate of P% per Example 8.B.4: A community of rabbits begins with an time period, the doubling time is approximately initial population of 100 and grows 7% per month. � ������ � 70 � (a) What is the approximate doubling time? This approximation works best for small growth rates and breaks down for growth rates over about 15%. (b) By what factor does the population increase in 18 months? (c) What is the population after 3 years? Doubling Time (Exponential Growth) Doubling Time (Exponential Growth) Example 8.B.5: A community of zombies doubles every 6 Example 8.B.6: The number of DMACC students doubles in hours. every 16 years. (a) What is the approximate rate (percent) of increase? (a) What is the approximate rate (percent) of increase? (b) If the population was 18,000 students in 2000, what will the population be in 2030? (b) By what factor does the population increase in 24 hours? (c) What is the population after one week? (c) By what factor did the population increase in in that period? 5
3/6/2012 Half-Life Time (Exponential Decay) Approximate Half-Life Time Formula (The Rule of 70) • After a time � , an exponentially decreasing quantity with For a quantity decaying exponentially at a rate of P% per a half-life time of � ���� decreases in size by a factor of time period, the half-life time is approximately � � ���� ⁄ � ���� � 70 � . The new value of the decreasing quantity is � � related to its initial value (at � � 0 ) by This approximation works best for small decay rates and � � ���� ⁄ ��� ����� � ������� ����� � 1 breaks down for decay rates over about 15%. 2 Whatever unit of time is used to measure your half-life period is the unit of time you should use for � . For example, if a radio isotope decays with a half-life of 5100 years, then � must be measured in years. Half-Life Time (Exponential Decay) Half-Life Time (Exponential Decay) Example 8.B.7: Carbon-14 is used to carbon-date decaying Example 8.B.8: You start with 10 pounds of compost. It remains, whether it be plant or animal. The half-life of takes 3 months to break down to 5 pounds. Carbon-14 is 5730 years. (a) Write an equation modeling the amount of compost. (a) Write an equation modeling the amount of Carbon-14 of an object. (b) How many pounds will their be after 1 year? (b) Using guess and check, if a bone has 10% of its original carbon-14 left, how old is the bone? (c) By what factor did the population decrease in in that period? 6
3/6/2012 Half-Life Time (Exponential Decay) Exact Formulas Example 8.B.9: Since 1900 the buying power of one dollar • For more precise work use these exact formulas. has decreased 3% per year. • For an exponentially growing quantity, the doubling time is log 2 (a) What is the approximate half-life time? � ������ � log�1 � �� • For an exponentially decreasing quantity, the doubling (b) How much buying power does a 1900 dollar approximately have time is today? log 2 � ���� � � log�1 � �� • In both cases � is the percent (as a decimal). (c) By what factor did the 1900 dollar decrease in in that period? Log is a mathematical function similar to a square root. I could teach you how to calculate it by hand and what it truly means but is easier if we just skip that and know it is a button on our calculator. Exact Formulas Exact Formulas Example 8.B.10: Calculate each of the following. Example 8.B.11: Calculate each of the following. (a) What is the approximate half-life time of a quantity that decays by (a) What is the approximate doubling time of a quantity that increases 7% per month? by 4% per year? (b) What is the exact half-life time of a quantity that decays by 7% per (b) What is the exact doubling time of a quantity that increases by 4% month? per year? 7
3/6/2012 Exponential Growth/ Decay Exponential Growth/ Decay Example 8.B.12: If you had an account that had a Example 8.B.13: Urban encroachment is causing the area compound interest rate of 5% per month, how long would it of forest to decline at a rate of 4.25% per year. take for your money to double? (a) What is the exact half-life time? log 2 log 2 Month Interest Balance � ���� � � log 1 � .0425 � 15.96 � ������ � log�1 � .05� � 14.206 0 1000.00 1 50.00 1050.00 2 52.50 1102.50 3 55.13 1157.63 4 57.88 1215.51 (a) If the forest in that local area started with 500,000 acres, how 5 60.78 1276.28 much will be left after 20 years? 6 63.81 1340.10 7 67.00 1407.10 8 70.36 1477.46 9 73.87 1551.33 10 77.57 1628.89 ��� ����� � 500,000 0.5 �� ��.�� ⁄ � 209,768.09 11 81.44 1710.34 12 85.52 1795.86 The amount has doubled between month 14 and 15. 13 89.79 1885.65 Also, notice that we started with month 0 . 14 94.28 1979.93 15 99.00 2078.93 Acres in Perspective • One square mile is 640 acres. • Roughly 209,768 acres. 8
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