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Applications of Exponential Growth & Decay

Explore exponential growth and decay models by predicting populations and caffeine levels using rates of change. Understand how to calculate and project values over time accurately.

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Applications of Exponential Growth & Decay

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  1. Applications of Exponential Growth & Decay

  2. Model: f(x) = a · (b)x a: the starting value b: rate of change (base) x: variable of time (exponent)

  3. Fold a piece of notebook paper 50 times & determine the number of layers of paper.

  4. Fold a piece of notebook paper 50 times & determine the number of layers of paper.

  5. Model: f(x) = a · (b)x a: the starting value b: rate of change (base) x: variable of time (exponent) Answer: f(50) = 1 · (2)50 = 1.126 x 1015 layers

  6. 1) The Populationof the U.S. was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

  7. Solution: Step 1) To obtain the base (rate of change) for exponential growth, add the growth rate to 100%. 100% + 8% = 108% Or 1.08 WHY??

  8. Step 2) Write the expression for the population x decades after 1990 (since 1990 is your starting year). f(x) = a · (b)x = 248,718,301 · (1.08)x

  9. Step 3 a.) Since the year 2010 is 2 decades (20 years) after 1990, substitute 2 for x. = 248,718,301 · (1.08)x = 248,718,301 · (1.08)2 = 290,105,026.3 ~290,100,000 people

  10. Step 3 b.) Since the year 2025is 3.5 decades (35 years) after 1990, substitute 3.5 for x. = 248,718,301 · (1.08)x = 248,718,301 · (1.08)3.5 = 325,604,866 ~325,600,000 people bonus opp: check the US's current population. Are the predictions on track?

  11. 2) EliminationThe rate at which caffeine is eliminated from the bloodstream of an adult is about 15% per hour. An adult drinks a soda, and the caffeine in his bloodstream usually reaches a peak level of 30 milligrams Predict the amount, to the nearest 10th of a milligram, of caffeine remaining 1 hour after the peak level & 4 hours after the peak level.

  12. Solution: Step 1) To obtain the base (rate of change) for exponential decay, subtract the rate of decay from 100%. 100% - 15% = 85% Or .85 WHY??

  13. Step 2) Write the expression for the caffeine level x hours after the peak level. f(x) = a · (b)x = 30 · (0.85)x

  14. Step 3 a.) Substitute 1 (1 hour) for x. = 30 · (0.85)x = 30 · (0.85)1 = 25.5 mg

  15. Step 3 b.) Substitute 4 (4 hours) for x. = 30 · (0.85)x = 30 · (0.85)4 = 15.7 mg bonus opp: how many hours until you're caffeine free?

  16. The birth rate for a colony of ants is 60%. The death rate is 46%. Assume that at the beginning of the observations the population was 650. • What will the population be in 5 years? 20 years?

  17. What is the difference between the birth and death rates? Is the population increasing or decreasing? Write an equation that could model this situation. f(x) = a · (b)x

  18. 4) The birth rate for a school of guppies is 56%. The death rate is 64%. Assume that at the beginning of the observations the population was 800. What will the population be in 12 years? 50 years?

  19. What is the difference between the birth and death rates? Is the population increasing or decreasing? Write an equation that could model this situation. f(x) = a · (b)x

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