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Exponential Growth & Decay Activity

Exponential Growth & Decay Activity. # of Ballots. # of Cuts. 0 1 2. Number of Ballots Number of Ballots after 1 cut Number of Ballots after 2 cuts. Each person needs a piece of paper: Cut your sheet of paper like Alejandra did Make a Table and then graph the data.

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Exponential Growth & Decay Activity

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  1. Exponential Growth & Decay Activity # of Ballots # of Cuts 0 1 2 Number of Ballots Number of Ballots after 1 cut Number of Ballots after 2 cuts Each person needs a piece of paper: Cut your sheet of paper like Alejandra did Make a Table and then graph the data

  2. Does your graph look like this? Why? What pattern do you notice in the table and the graph?

  3. Try to write an equation that models your data. If you started with 1 Ballot… Number of Ballots= 1 x 2 x 2 x 2 x 2 x . . . # of trials completed B = 12x y = (start amt) (factor of change)x

  4. Thirsty? Suppose I have 500 mL of water, and then pour half of that water into cup #1. How much water is in cup #1? Next I pour half of the water from cup #2 into cup #3. How much water is in cup #2? 250 125 62.5 If this process continues, how much water will be in cup #9?

  5. Thirsty? Can you write an equation for the data in the table? w = 500 x 0.5 x 0.5 x 0.5 x … w = 500(0.5c)

  6. How can you tell if a table represents an exponential relationship? x 0.5 x 0.5 x 0.5 exponential – the data changes by a constant factor The y-values represent a geometric sequence.

  7. Exponential Growth and Decay • Standard form of an exponential function is y = abx • a = initial amount • b = growth/decay factor • x = # of time intervals

  8. What’s the difference between growth and decay? y = 20(1.4)x • Put this function into your calculator and create a table. • What is the initial amount in the equation and where is that value in the table? • What factor are we multiplying by from one term to the next? • So the data is growing at a rate of 40%. • The growth factor is 1.4

  9. What’s the difference between growth and decay? y = 20(0.6)x • Put this function into your calculator and create a table. • What was the initial amount? • What factor are we multiplying by from one term to the next? • The data is decaying at a rate of 40% (100% - 40% = 60%). • The decay factor is 0.6.

  10. Exponential Growth and Decay Function • y = abx • a = initial amount • b = growth/decay factor • x = # of time intervals when b > 1 the equation is a growth function when 0 < b < 1 the equation is a decay function

  11. Growth and Decay Factor • For problems involving a growth rate: • Convert the percent to a decimal. • 2. Add the decimal to 1 to find b. • For problems involving a decay rate: • Convert the percent to a decimal. • 2. Subtract the decimal from 1 to find b. b = 1 ± r (where r is your growth/decay rate) For example…

  12. Finding “b” . . . If the data increases by 20%, then your factor is 1.2 100% + 20% = 120% = 1.2 20% is your growth rate & 1.2 is your growth factor If the data decreases by 20%, then your factor is 0.8 100% - 20% = 80% = 0.8 20% is your decay rate & 0.8 is your decay factor When a problem says an amount is doubling, what should we use for b? If it says the amount is tripling, what should we use for b?

  13. Exponential Growth and Decay Examples 1) A rabbit population triples every year. If I start with 4 rabbits, how many rabbits will I have in 7 years? R = 4 (37) = 8748 rabbits • The new drug to treat SARS has a half-life of 3 hours. If 50cc is initially administered, how much will still be in your system 24 hours later? D = 50 (.58)  0.20 cc

  14. 3) Evans town’s population in the year 1910 was 4,210. If we know that the population grew at a rate of 4.5% per year, what was the population in the year 1936? P = 4210 (1.04526)  13,222 people 4) $500 is invested in a bank for 18 years. If the money made 12.5% each year, what was the value when the account was closed? M = 500 (1.12518)  $4165.96

  15. In 1998, the population of humpback whales began decreasing at a rate of 7.4% per year. If there were 176,534 whales in 1998, how many should they expect in 2015? W = 176534 (0.92617)  47,777 whales • Ms. Hirsch bought a Honda Accord worth $25,000. The car’s value will depreciate by about 5.5% every year. How much will the car be worth in 6 years when the loan is paid off? C = 25000 (0.9456)  $17, 804.54

  16. Working backwards… 7. The population of white sea otters is declining at a rate of 2.5% each year. If the current population is 1.9 million, how many white sea otters were there 5 years ago? a  2,156,406 1.9 = a (0.9755) 8. Suppose you have 30,000 bacteria in a petri dish. You know that the culture doubles every hour. How many bacteria did you have 5 hours ago? 30000 = a (25) a = 937.5 bacteria

  17. Working backwards… 9. Ten years ago, the Smiths bought a house for $96,000. Their home is now worth $125,000. Assuming there has been a steady growth rate, what was the annual rate of appreciation? r  2.7 % 125000 = 96000 (1+r)10 10. Mr. Adams bought his car for $28,000 five years ago. If it is now worth $18,000, what was the annual rate of depreciation? 18000 = 28000 (1-r)5 r  8.5 %

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