Exponential Decay

Examples, videos, and solutions to help Algebra I students learn how to describe and analyze exponential decay models; they recognize that in a formula that models exponential decay, the growth factor b is less than 1; or, equivalently, when bis greater than 1, exponential formulas with negative exponents could also be used to model decay.

New York State Common Core Math Algebra I, Module 3, Lesson 7

Worksheets for Algebra I, Module 3, Lesson 7 (Teacher Version)
Worksheets for Algebra I, Module 3, Lesson 7 (Student Version)

Lesson Summary
The explicit formula f(t) = ab^t models exponential decay, where a represents the initial value of the sequence, b < 1 represents the growth factor (or decay factor) per unit of time, and t represents units of time.

Exercises

1. Identify the initial value in each formula below, and state whether the formula models exponential growth or exponential decay. Justify your responses.
2. If a person takes a given dosage (d) of a particular medication, then the formula f(t) = d(0.8)^t represents the concentration of the medication in the bloodstream hours later. If Charlotte takes 200 mg of the medication at 6:00 a.m., how much remains in her bloodstream at 10:00 a.m.? How long does it take for the concentration to drop below 1 mg?

Exit Ticket
A huge ping-pong tournament is held in Beijing, with 65.536 participants at the start of the tournament. Each round of the tournament eliminates half the participants.
a. If p(r) represents the number of participants remaining after r rounds of play, write a formula to model the number of participants remaining.
b. Use your model to determine how many participants remain after 10 rounds of play.
c. How many rounds of play will it take to determine the champion ping-pong player?

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Problem Set Sample Solutions

1. From 2000 to 2013, the value of the U.S. dollar has been shrinking. The value of the U.S. dollar over time v(t) can be modeled by the following formula:
   v(t) = 1.36(0.9758)^t, where t is the number of years since 2000.
   a. How much was a dollar worth in the year 2005?
   b. Graph the points(t, v(t)), for integer values of 0 ≤ t ≤ 14.
   c. Estimate the year in which the value of the dollar fell below $1.00.
2. A construction company purchased some equipment costing $300,000. The value of the equipment depreciates (decreases) at a rate of 14 % per year.
   a. Write a formula that models the value of the equipment each year.
   b. What is the value of the equipment after 9 years?
   c. Graph the points(t, v(t)), for integer values of 0 ≤ t ≤ 15
   d. Estimate when the equipment will have a value of $50,000.
3. The number of newly reported cases of HIV (in thousands) in the United States from 2000 to 2010 can be modeled by the following formula:
   f(t) = 41(0.9842)^t, where t is the number of years after 2000.
   a. Identify the growth factor.
   b. Calculate the estimated number of new HIV cases reported in 2004.
   c. Graph the points(t, v(t)), for integer values of 0 ≤ t ≤ 10
   d. During what year did the number of newly reported HIV cases drop below 36,000?
   
4. Doug drank a soda with 130 mg of caffeine. Each hour, the caffeine in the body diminishes by about 12%.
   a. Write formula to model the amount of caffeine remaining in Doug’s system each hour.
   b. How much caffeine remains in Doug’s system after 2 hours?
   c. How long will it take for the level of caffeine in Doug’s system to drop below 50 mg?
5. 64 teams participate in a softball tournament in which half the teams are eliminated after each round of play.
   a. Write a formula to model the number of teams remaining after any given round of play.
   b. How many teams remain in play after 3 rounds?
   c. How many rounds of play will it take to determine which team wins the tournament?
6. Sam bought a used car for $8,000. He boasted that he got a great deal since the value of the car two years ago (when it was new) was $15,000. His friend, Derek, was skeptical, stating that the value of a car typically depreciates about 25% per year, so Sam got a bad deal.
   a. Use Derek’s logic to write a formula for the value of Sam’s car. Use t for the total age of the car in years.
   b. Who is right, Sam or Derek?

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