# Exponential Decay: Examples & Definition

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• 0:00 What Is Exponential Decay?
• 0:20 Example
• 3:00 The Math
• 4:15 Exponential Decay And e
• 5:20 Lesson Summary

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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.

## What Is Exponential Decay?

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.

## Example

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.

## The Math

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.

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