Unit 9, Lesson 5
Understanding Decay
Integrated Math 1
5.1
Warm-up
What do you notice? What do you wonder?
Table A
| 0 | 2 |
| 1 | |
| 2 | 5 |
| 3 | |
| 4 | 8 |
Table B
| 0 | 2 |
| 1 | 3 |
| 2 | |
| 3 | |
| 4 |
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What do you notice? What do you wonder?
Table A
| 0 | 2 |
| 1 | |
| 2 | 5 |
| 3 | |
| 4 | 8 |
Table B
| 0 | 2 |
| 1 | 3 |
| 2 | |
| 3 | |
| 4 |
Every year after a new car is purchased, it loses of its value. Let’s say that a new car costs \$18,000.
If the car loses of its value every year, how much is the car still worth?
Pause here for a whole-group discussion.
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 | |
Sometimes a quantity grows by the same factor at regular intervals. For example, a population might double every year. Sometimes a quantity decreases by the same factor at regular intervals. For example, a car might lose one third of its value every year.
Let's look at a situation in which the quantity decreases by the same factor at regular intervals. Suppose a bacteria population starts at 100,000, and of the population dies each day. The population one day later is , which can be written as . The population after one day is of 100,000 or 75,000. The population after two days is . Here are some further values for the bacteria population:
| number of days |
bacteria population |
|---|---|
| 0 | 100,000 |
| 1 | 75,000 (or ) |
| 2 | 56,250 (or , or ) |
| 3 | about 42,188 (or , or ) |
In general, days after the bacteria population was 100,000, the population is given by the equation: with one factor of for each day.
Situations with quantities that decrease exponentially are described with the term exponential decay. The multiplier ( in this case) is still called the growth factor, though sometimes people call it the decay factor instead.