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NY Regents Exam - Integrated Algebra: Help and Review24 chapters | 260 lessons

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Lesson Transcript

Instructor:
*Kimberlee Davison*

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

Exponential decay occurs when a population decreases at a consistent rate over time. In this lesson, you will learn what makes exponential decay unique.

When a population or group of something is declining, and the amount that decreases is proportional to the size of the population, it's called **exponential decay**. In exponential decay, the total value decreases but the proportion that leaves remains constant over time.

Suppose that you had a bowl full of jelly beans on the coffee table in your apartment. You notice that each day, the bowl looks a little more empty - in spite of the fact that you are on a diet and have completely sworn off jelly beans.

Being a little obsessive compulsive, you count the number of jelly beans left in the bowl each night before you go to bed. It looks like this:

Table One:

Day 1: 890 beans

Day 2: 801 beans

Day 3: 721 beans

Day 4: 649 beans

Day 5: 584 beans

So far, that doesn't look too interesting. The beans are gradually decreasing but fewer each day. Maybe whoever is stealing your jelly beans is getting sick of them, and by the time you are off your diet next month, there will be some left for you. In order to test that hypothesis, you look a little more closely at the pattern in the number of beans missing each day:

Table Two:

Day 2: 89 beans missing

Day 3: 80 beans missing

Day 4: 72 beans missing

Day 5: 65 beans missing

The pattern doesn't look all that meaningful until you compare table two with table one. Suddenly, you notice that each day, the number of beans that goes missing is about 10% of the beans that were in the jar the previous day. Eighty-nine beans go missing out of 890, 80 beans go missing out of 801, and so forth. To test your theory, you predict that 10% of the beans available tonight (day five), or 58 beans, will go missing before your next count. Sure enough, you discover that there are 526 beans in the jar on day six. It appears that your thief is just as obsessive compulsive as you are.

What is happening here is 'exponential decay' because the rate of decrease stays consistent from day-to-day. No matter how big your 'population' of jelly beans is on any given day, about 10% will vanish by the next day.

If you graph the jelly bean count over time, the curve seems to decrease quickly and then level off.

From the graph, you can see that you'll run out of jelly beans after about two months. Technically, with exponential decay, the population doesn't ever quite reach zero - it just gets really, really close to zero over time (there is an asymptote at *y* = 0). Of course, if you limit your thief to eating whole numbers of jelly beans, then it isn't quite that simple. At some point, your thief is going to eat the last remaining jelly bean, rather than just 90% of a jelly bean.

You may want to create an equation, or function, that gives you the number of jelly beans you'll have after so many days.

Each day, you have left 90%, or 0.90, of the beans from the prior day. So, after one day (on day 2), you have 890 * 0.90 beans. The next day you have 90% of that, or (890 * 0.90) * 0.90.

The pattern continues. After five days, you have 890 * 0.90 * 0.90 * 0.90 * 0.90 * 0.90 beans, which could be written more concisely as: 890 * 0.90^5 (The ^ symbol means to raise 0.90 to the fifth power.)

After *t* days, you would have 890 * 0.90^*t* beans.

In general, exponential decay always looks like:

(Amount after *t* amount of time) = (Starting quantity) * (percentage) ^*t* or

*A* = *N* * *b*^*t*, where *b* is the base, and must be less than 1, and *N* is the amount you start with.

For the moment, we'll pretend that we can consider fractional days; you might be able to count the beans after 3.5 days, for example, and that the beans are also vanishing at a consistent rate (exponential decay) within days.

You might be saying, 'Wait! That isn't what I learned in math class. In my class, the teacher always uses *e* when we talk about exponential decay.' Don't worry. We're getting to that, and there is a link to what we've done so far.

Most often, you will see the formula for exponential decay written this way, or some variation on this:

Rather than,

Getting from one formula to the other is just substitution and a little algebra. In formula two *b* is the amount left after each period of time - a day, in our jelly bean example. In formula one, *e*^-*r* replaces the *b.* The *e*, or **Euler's Number**, is simply a constant that is close to 2.71828 but with decimal places that go on forever (*e* is irrational). *r* is usually called the rate, and it is related to the percent that the population declines each unit of time, for instance, a day.

Generally equation one is easier to solve than equation two, so it's used more often.

Let's review. **Exponential decay** occurs when a population declines at a consistent rate. No matter how many are in the population at some point in time, the percent that leave the population in the next period of time will be consistent. In general, exponential decay always looks like this: (Amount after *t* amount of time) = (Starting quantity) * (percentage) ^*t*, or this: *A* = *N* * *b*^*t* .

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NY Regents Exam - Integrated Algebra: Help and Review24 chapters | 260 lessons

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