Exponents and Exponential Functions

Lesson 21

Objective

 Identify features of exponential decay in real-world problems.

Common Core Standards

Core Standards

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  • F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01 12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

  • F.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Foundational Standards

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  • 8.F.B.4

Criteria for Success

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  1. Determine if a real-world situation is exponential growth or exponential decay.
  2. Identify the rate of decay in a real-world situation. 
  3. Write and evaluate exponential decay functions for applications.

Anchor Problems

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Problem 1

Malik bought a new car for $15,000. As he drove it off the lot, his best friend, Will, told him that the car’s value just dropped by 15% and that it would continue to depreciate 15% of its current value each year. 

  1. If the car’s value is now $12,750 (according to Will), what will its value be after 5 years? Complete the table below to determine the car’s value after each of the next five years. Round each value to the nearest cent. 
# of years, t, since driving the car off the lot Car value after t years 15% depreciation of current value Car value minus 15% depreciation
0 $12,750.00 $1,912.50 $10,837.50
1      
2      
3      
4      
5      
  1. Write an exponential function to model the value of Malik’s car $$t$$ years after driving it off the lot. 
  2. Use the function from part (b) to determine the value of Malik’s car 5 years after its purchase. Round your answer to the nearest cent. Compare the value with the value in the table. Are they the same?
  3. Use the function from part (b) to determine the value of Malik’s car 7 years after its purchase. Round your answer to the nearest cent. 

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 7Example 1

Algebra I > Module 3 > Topic A > Lesson 7 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

Problem 2

Every day Brian takes 20 mg of a drug that helps with his allergies. His doctor tells him that each hour the amount of drug in his bloodstream decreases by 20%. 

  1. Construct an exponential function in the form $${f(t)=ab^t}$$, for constants $$a$$ and $$b$$, that gives the quantity of the drug, in milligrams, that remains in Brian’s bloodstream $$t$$ hours after he takes the medication. 
  2. How much of the drug remains one day after taking it?
  3. Do you expect the percentage of the dose that leaves the bloodstream in the first half hour to be more than or less than 20%? What percentage is it?
  4. How much of the drug remains one minute after taking it?

Guiding Questions

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References

Illustrative Mathematics Allergy Medication

Allergy Medication, accessed on May 17, 2018, 1:35 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by The Match Foundation, Inc.

Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

Target Task

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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.

  1. If $${p(r)}$$ represents the number of participants remaining after $$r$$ rounds of play, write a function to model the number of participants remaining. 
  2. Use your model to determine how many participants remain after 10 rounds of play. 
  3. How many rounds of play will it take to determine the champion ping-pong player?

References