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Algebra 1

5.5.2 Exponential Decay

Algebra 15.5.2 Exponential Decay

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Activity

Every year after a new car is purchased, it loses 1 3 1 3 of its value. Another way to think about this is that the car retains 2 3 2 3 of its value each year. Let’s say that a new car’s value is $18,000.

1. A buyer worries that the car will be worth nothing in three years. Do you agree? Be prepared to show your reasoning.

2. Write an expression to show how to find the value of the car for each year listed in the table.

Year Value of Car (Dollars)
0 18,000
1  
2  
3  
6  
t t  

3. Write an equation relating the value of the car in dollars, v v , to the number of years, t t .

4. Use your equation to find v v when t t is 0. What does this value of v v mean in this situation?

5. A different car loses value at a different rate. The value of this different car in dollars, d d , after t t years can be represented by the equation d = 10 , 000 · ( 4 5 ) t d = 10 , 000 · ( 4 5 ) t . Explain what the numbers 10,000 and 4 5 4 5 mean in this situation.

Are you ready for more?

Extending Your Thinking

Start with an equilateral triangle with area 1 square unit, divide it into 4 congruent pieces as in the figure, and remove the middle one. Then, repeat this process with each of the remaining pieces. Repeat this process over and over for the remaining pieces. The figure shows the first two steps of this construction.

Three blue triangles show the stages of forming a Sierpinski triangle: starting with a solid triangle, then removing the center, and finally removing the centers of the remaining triangles, creating a recursive pattern.

What fraction of the area is removed each time? How much area is removed after the nth step? Use a calculator to find out how much area remains in the triangle after 50 such steps have been taken.

Self Check

Which of the following could represent a population of bacteria, b , that decreases at a rate of 1 8 every hour, h ? Consider an initial amount of 100 bacteria.
  1. b = 100 ( 1 8 ) h
  2. b = 100 ( 7 8 ) h
  3. b = 100 ( 1 8 ) h
  4. b = 100 ( 7 8 ) h

Additional Resources

Analyzing Exponential Decay

At the beginning of April, a colony of ants had a population of 5,000.

  1. The colony decreases by 1 5 1 5 during April. Write an expression that represents the ant population at the end of April.
  2. During May, the colony decreases again by 1 5 1 5 of its size. Write an expression that represents the ant population at the end of May.
  3. The colony continues to decrease by 1 5 1 5 of its size each month. Write an expression that represents the ant population after 6 months.
  4. Write an equation to represent the population of ants, p p , after m m months.

When you are analyzing exponential decay and are given the fraction or percent by which it decreases, you need to remember that one minus that fraction or percent is the growth rate. In the case of the ant colony, because it is decreasing by 1 5 1 5 , that means that the growth factor is 1 1 5 1 1 5 or 4 5 4 5 .

Part 1 asks for an expression to represent the ant population at the end of April. There are several options:

  • 5000 ( 1 5 · 5000 ) 5000 ( 1 5 · 5000 )
  • 5000 · 4 5 5000 · 4 5
  • 4000 4000

Each line is the simplification of the line above it. Part 2 has a few more options.

Part 2 is asking you to find 4 5 4 5 of April’s total. There are several expressions to represent that:

  • 5000 · 4 5 · 4 5 5000 · 4 5 · 4 5
  • 4000 · 4 5 4000 · 4 5
  • 4000 ( 1 5 · 4000 ) 4000 ( 1 5 · 4000 )
  • 3200 3200

Basically, you take the answer from April and multiply by 4 5 4 5 again since the growth factor is 4 5 4 5 . You could also look at it as taking the initial value times 4 5 4 5 twice.

Part 3 asks you to extend the pattern to 6 months after the initial population was measured. After the 2nd month, you multiplied the initial value by the growth factor twice, so for this situation you would need to multiply the initial value by the growth factor 6 times, or raise it to the sixth power. The possible answers would be 5000 · 4 5 · 4 5 · 4 5 · 4 5 · 4 5 · 4 5 5000 · 4 5 · 4 5 · 4 5 · 4 5 · 4 5 · 4 5 or 5000 · ( 4 5 ) 6 5000 · ( 4 5 ) 6 , which means that after 6 months the population would be about 1,311.

Part 4 wants the general equation to represent the situation. If you use the pattern from part 3, you can see that it can be written in the exponential form y = a b x y = a b x , where a a is the initial value and b b is the growth factor. Using the variables suggested, the equation would be p = 5000 ( 4 5 ) m p = 5000 ( 4 5 ) m .

Try it

Try It: Analyzing Exponential Decay

A doctor prescribes 125 milligrams of a therapeutic drug that decreases in effectiveness by 3 10 3 10 each hour.

a. Write an expression to represent how much is left after 1 hour.

b. Write an expression to represent how much is left after 3 hours.

c. Write an equation that represents the amount of drug, d d , left after h h hours.

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