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Warm Up

Warm Up. Exponential Investigation…. Take a sheet of notebook paper and fold it in half. Fold it twice and now 3 times. It's about as thick as a finger nail. Continue folding if you can.

jennifer-lane
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Warm Up

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  1. Warm Up

  2. Exponential Investigation… • Take a sheet of notebook paper and fold it in half. • Fold it twice and now 3 times. It's about as thick as a finger nail. • Continue folding if you can. • At 7 folds it is as thick as a notebook. Unfortunately, it isn't possible to do so more than about 7 times. • If you would have been able to fold it 10 times, it would be as thick as the width of your hand. • At seventeen folds it would be taller than your average house.! • 3 more folds and the paper is a quarter way up the Sears tower! • Ten more folds and it has crossed the outer limits of the atmosphere. Another twenty and it has reached the sun from the earth. At sixty folds it has the diameter of the solar system! At 100 folds it has the radius of the universe!

  3. Exponential Growth and decay

  4. Why Exponential growth & decay can be used to model a number of real-world situations, such as population growth of bacteria & the elimination of medicine from the bloodstream. It is also used in business to calculate the value of investments.

  5. Sample Graphs y=3x y=(1/3)x

  6. Discussion Sally starts a chain email that requires each recipient to pass it on to five new people after 24 hours else they be cursed with a lifetime of bad-hair days. Sally sends the email to three of her friends. If everyone who gets the email obeys and sends it on to five people, how many people will receive the email on the 15th day after it started? Initial amount: 3 Growth Factor: 5 Length of Growth: 15 Final amount: y

  7. Exponential Growth Model y = final amount a = initial amount t = time r = % of increase 1+r = growth factor

  8. Notes If the growth is given as a rate in percent, it must be converted to the growth factor by first changing it to a decimal, then adding 1. Example: Growth rate: 32% Growth factor: 1.32 Growth Rate b = 1 + r

  9. Discussion Two grey squirrels stow away on a freight ship heading for Greenland. When they arrive, they take to their new environment and begin to thrive. Their population grows at a rate of 45% per month. How many squirrels will there be after one and a half years? Initial amount: 2 Growth rate: 45% Growth factor: 1.45 Length of Growth: 18 months Final amount: y

  10. In 1990, the cost of tuition at a state university was $4300. Tuition increases 4% each year. • Write a model the gives the tuition y (in dollars) t years after 1990 • What is the growth factor? • How much did it cost to attend college in 2000? In 2007? When you will enter college?

  11. You inherited some land in 1960 worth $30,000. The value of the property increases 5% per year. What it be in 2016? a= initial amount, r= interest rate x = time in years $344,021

  12. Exponential Decay • A quantity decays exponentially when its decrease is proportional to what is already there.

  13. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • If you don't understand exponential decay -- just pour yourself a coke over ice and watch what happens. • The rate at which the foam on a coke shrinks models exponential decay.

  14. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • To understand exponential decay, think of the coke foam as a collection of bubbles. • In one time period, half of the bubbles will pop, and the height of the foam will drop by half. • In a second time period, half of the remaining half will pop, leaving only a quarter of the original foam.

  15. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • In a third time period, another eighth of the original bubbles (half of the remaining quarter) will pop, and the height will be an eighth of where it started. • As bubbles pop, there are fewer left to pop, which is why the decay slows.

  16. Notes Exponential Decay y: Final amount a: Initial amount r: decay rate t: Length of decay (days, minutes, etc.)

  17. Notes If the decay is given as a rate in percent, it must be converted to the decay factor by changing it to a decimal and subtracting it from 1. Example: Decay Rate: 12% Decay Factor: 0.88 Decay Rate b = 1 – r

  18. Ex. • You bought a new car for $22,000. The car depreciates 12.5% per year. • What is the value of the car after 3 years? • When will the car have a value of $8000? a= initial amount, r= interest rate t = time in years $14738 7.65 years

  19. Discussion You purchase a brand new 2007 BMW for $45,000.00. Each year the value of a car depreciates about 15%. How much is your BMW worth in the year 2020? Initial amount: 45,000 Decay Rate: 15% Decay Factor: 0.85 Length of Decay: 13 Final amount: y

  20. Word Problem A new all-terrain vehicle costs $800. The value decreases by 10% each year. Write an exponential decay model for the value of the ATV (in dollars) after t years. Estimate the value after 5 years. y = 800(0.90)t After 5 years, $472.39

  21. Homework • Worksheet

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