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High School Algebra I: Help and Review25 chapters | 292 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 growth is growth that increases at a consistent rate, and it is a common occurrence in everyday life. In this lesson, learn about exponential growth and some of its real-world applications.

Some things grow at a consistent rate. Money or the descendants of mating rabbits, for example, can grow faster and faster as the total number itself gets bigger. When growth becomes more rapid in relation to the growing total number, then it is **exponential**.

Exponential growth is extremely powerful. One of the most important features of exponential growth is that, while it starts off slowly, it can result in enormous quantities fairly quickly - often in a way that is shocking.

There is a legend in which a wise man, who was promised an award by a king, asks the ruler to reward him by placing one grain of rice on the first square of a chessboard, two grains on the second square, four grains on the third and so forth. Every square was to have double the number of grains as the previous square. The king granted his request but soon realized that the rice required to fill the chessboard was more than existed in the entire kingdom and would cost him all of his assets.

The number of grains on any square reflects the following rule, or formula:

In this formula, *k* is the number of the square and *N* is the number of grains of rice on that square.

- If
*k*= 1 (the first square), then*N*= 2^0, which equals 1. - If
*k*= 5 (the fifth square), then*N*= 2^4, which equals 16.

This is exponential growth because the exponent, or power, increases as we go from square to square.

There are a variety of examples of exponential growth as it applies to the real world. For example, a man is believed to have brought 24 rabbits to Australia in the 1800s so that he could hunt them; however, rabbits have no natural predators in Australia, and so the population grew out of control. Within ten years, so many rabbits had descended from these 24 first rabbits that millions could be killed without making a dent in the population.

The real secret to exponential growth is this - not only do rabbits have children, but their children have children, as do their children's children. The new growth increases just as fast as the growth you started with. There is nothing to slow the growth down or bring it to a halt.

When graphed, exponential growth always looks like it is starting off slowly and then rapidly becomes steeper:

It's a lot like spreading gossip about your ex: you might only tell your two best friends that he cries during chick flicks, but each of them tells a couple others, and pretty soon there is no one in the Western Hemisphere that doesn't know his secret - thanks to the power of exponential growth.

Let's take a look at another example: suppose you have bacteria in a Petri dish. Maybe each bacteria splits in two; each of those splits in two and so forth. Since the bacteria are tiny and the dish is much bigger in comparison, there is nothing to stop the growth for a very long time. Like the rabbits in Australia, there is no force to stop the growth. In other real-world situations, of course, there might predators or limits to food sources so that growth can't go on forever at the same rate but, instead, slows down. In this case, you have **logistic growth**. It starts off looking exponential but eventually levels out.

Not all exponential growth is the same. The rice in the chessboard story doubled from square to square. It would still be exponential growth if they tripled instead, but the world would run out of rice faster. Or the rule might be that each square only has ten percent more rice than the previous one. In this case, the growth is slower, but it is still exponential.

One example of slow exponential growth is money in a savings account. Suppose you are getting one percent interest each year, paid at the end of the year (which is much simpler than what really happens). This means that you get one dollar for every $100 you have in the account. Or, you get one cent for every 100 cents. If you start off with $452.10, then the interest you get at the end of the year is $4.52 - very simple. But, the second year you get interest not only on your original $452.10, but also on the $4.52 interest you earned in year one. So, the amount that gets added to your account is slightly bigger in year two and increases year after year. The interest rate may be small, but the growth of your funds is still exponential - the amount earned in interest increases year to year even though the rate remains the same.

When growth becomes more rapid in relation to the growing total number, then it is **exponential**. With exponential growth, the actual quantity added over time gradually increases. There is no force that slows down or stops the growth to any noticeable degree.

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High School Algebra I: Help and Review25 chapters | 292 lessons

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