Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 1 Exponential decay: finding the multiplier In general, the equations for exponential decay look like: y = a · bx. NEXT = NOW · b, starting from a.

In decay problems, the multiplier b is always between 0 and 1. Today’s lesson is about different ways to find the b value from different kinds of decay descriptions.

Method 1: Thinking about what fraction or decimal is kept In last night’s homework there was a problem (February 2, problem 4) about removing popcorn kernels. The problem indicated that

3

1 of the kernels were removed at each stage. To find the multiplier, we had to think about what fraction of the kernels was kept at each stage. Since

3

2 of the kernels were kept, the multiplier in that problem was b =

3

2 .

Here’s a rule you can follow in general:

b = the portion that’s kept, as a fraction or a decimal

Method 2: Using the fraction or decimal that has been removed In the popcorn problem, what’s the arithmetic that could be used to get from the fraction

3

1 that’s removed to the fraction

3

2 that’s kept? It’s subtraction from 1: 1 – 3

2 = 3

1 . So if you have a fraction or decimal describing how much has been removed, here’s a rule:

b = 1 – (the portion that’s been removed, as a fraction or decimal)

Method 3: Thinking about what percent is kept Suppose a problem involved a decrease by 20%. That would mean that 80% is kept. The multiplier is the percent kept, changed to a decimal, b = 0.8. Here’s a rule you can follow in general:

b = the percent that’s kept, converted to a decimal

Method 4: Using the percent that has been removed Suppose a problem involved a decrease by 20%. You could convert the 20% to a decimal, 0.2, then get the multiplier by subtracting from 1: b = 1 – 0.2 = 0.8. So if you have a percentage of decrease (that is, the percent that’s removed), here’s a rule:

b = 1 – (the percentage of decrease, changed to a decimal)

Comparison to exponential growth Previously we studied finding the multiplier for exponential growth. When growth was described using a percentage of increase the rule was

b = 1 + (the percentage of increase, changed to a decimal)

Note that the rule for exponential decay is almost the same, but involves – instead of +.

Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 2 Practice: finding multipliers 1. Below are various descriptions of exponential decay. For each, find the value of the

multiplier b. Examples are shown to get you started.

Description of decay Multiplier keep 94% b = 0.94

decrease by 6% b = 1 – 0.06 = 0.94

decrease by 35%

keep 35%

decrease by 5

1

keep 5

1

decrease by 10

7

keep 10

7

decrease by 2.7% 2. Here the multipliers are given. Fill in the blanks in the descriptions of the decay.

Description of decay Multiplier keep ____% b = 0.95

decrease by ____% b = 0.95

keep ____ [fill in with a fraction] b = 0.8

decrease by ____ [fill in with a fraction] b = 0.8

decrease by ____% b = 0.9817

3. Do the same as above, but growth and decay changes are mixed together here.

Description of change Multiplier decrease by 8%

increase by 8%

increase by 10%

decrease by 10%

___________ by ____% b = 1.14

___________ by ____% b = 0.91

increase by 3

2

decrease by 3

2

Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 3 Summary: how to set up an exponential model Generally, in exponential growth and decay problems, use equations that looks like: y = a · bx. NEXT = NOW · b, starting from a.

Here are the many ways we’ve studied to find the multiplier. • If you’re given a number that’s used for repeated multiplication, b = that number.

[example: b = 3 in a problem that involves tripling at each stage] • If you’re told what fraction or decimal is kept, b = that fraction or decimal. • If you’re given a growth rate as a fraction: b = 1 + that fraction. • If you’re given a decay rate as a fraction: b = 1 – that fraction. • If you’re given a growth rate as a percent: b = 1 + (growth % changed to decimal). • If you’re given a decay rate as a percent: b = 1 – (growth % changed to decimal).

Problems 4. The table shows the decrease in the number of

VCR’s sold each year by an electronics store chain. a. Why do you think VCR’s sales would be

decreasing as the years progress?

b. The number of VCR’s sold is decreasing by the same percentage each year. Identify the percentage of decrease, the percentage that’s kept, and the multiplier.

____% decrease

____% kept multiplier: b = ______

c. Write a y = ··· equation that fits the table.

d. Write a NOW-NEXT description that fits the table.

e. Assuming the pattern continues, fill in the next two rows in the table.

f. Assuming the pattern continues, how many VCR’s would be sold this year (2009)?

Find a way to get the answer without figuring out all the in-between years.

Years since 2000 Number of VCR’s sold 0 1,000,000 1 950,000 2 902,500 3 857,375

Years since 2000 Number of VCR’s sold 4 5

Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 4 5. The values of expensive products like automobiles depreciate from year to year. One

common method for calculating the depreciation of automobile values assumes that a car loses 20% of its value each year.

For this problem, suppose a new pickup truck costs $20,000, and the truck’s value depreciates by 20% each year.

a. The value of the truck one year later will be $16,000. Show the calculation that leads to this answer.

b. Write a NOW-NEXT rule that can be used to calculate the value of the truck from one year to the next.

c. Write a y = ··· rule for the truck’s value in any year.

d. Using an input-output table, estimate the time when the truck’s value is only $1,000.

(You can use your calculator to make the table, but copy down the part of the table that you use to get your answer.)

e. On graph paper: Make a graph of the truck’s value, from the time the time the truck is bought until the time that the truck is worth only $1,000.

Circle the points on the graph that represent the answers to part a and part d.

Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 5 6. A store receives a shipment of 1000 greeting cards. Each day, the store sells 2.5% of its stock

of cards. Let x = the number of days passed; y = the number of cards remaining in the store. a. Write a NOW-NEXT rule for the store’s supply of greeting cards.

b. Write a y = ··· rule for the store’s supply of greeting cards.

c. After 15 days, how many cards will remain in the store?

d. The store manager wants to order a new shipment when the remaining supply is 200 cards. After how many days will this happen? Show or tell how you get your answer.

7. A rock band gives a really bad concert. At the start of the concert, there are 2000 people in

the audience. After each song, 15 of the people in the audience leave.

Let x = the number of songs played; y = the number of people remaining in the audience.

a. Write a NOW-NEXT rule for the size of the audience.

b. Write a y = ··· rule for the size of the audience.

c. On graph paper: Make an input-output table and a graph.

Now use your table and graph to help answer these questions. d. After 10 songs, how many people were left in the audience?

e. The band stopped playing as soon as there were less than 200 people remaining in the audience. How many songs were played?

Name Algebra 1B notes and problems February 5, 2009 “Exponential decay: finding the multiplier” page 6 8. In the year 2000, the population of a town was 40,000 people. Suppose that the town’s

population has been declining by 1.5% each year, and that this rate of decrease will continue in the future.

Let x = the number of years since 2000, y = the town’s population. a. Write a NOW-NEXT rule for the town’s population.

b. Write a y = ··· rule for the town’s population.

c. Make a table showing the population every 5 years from 2000 through 2020.

d. What would be the town’s population be this year (2009)?