### 4.1 Exponential Growth

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• Math Help

In Example 1, you found a formula for the exponential growth of a bacteria culture. While this model is helpful for determining the number of bacteria after an elapsed time, it may be difficult to get a sense of the speed of the rate of exponential growth. Even a graph of the model might not be enough to communicate this. (You will learn more about graphing exponential growth later in Section 4.1.) The video below showing the growth of a second bacteria culture, however, is a powerful demonstration of the rate of exponential growth. Before watching the video, note that

• the initial population is one.
• the growth rate is 100% every 20 minutes.
• the video shows the growth during a 12-hour period, but the video is about 16 seconds in length.

In the formula for exponential growth, , the expression is sometimes referred to as the growth factor. To see why, let P = 1 and r = 300% = 3. Then evaluate for several values of n, where n is the time in years.

After the initial value, each value is 4 times the previous value. So, when the growth factor is 4, the value quadruples every year. Knowing that (1 + r) is the growth factor can help you answer questions such as, "What annual rate of growth will result in an amount tripling every year?" or "What annual rate of growth will result in an amount doubling every two years?" To answer the first question, you would need to find a value of r so that 1 + r = 3, which is a growth factor of 3 (tripling the value every year). The second question is a little more difficult to answer. See the Math Help for Example 6 in Section 4.1 to see one way to get the answer.

• Consumer Suggestion

Did you know that your cell phone has more bacteria per inch than a trash can or toilet seat? A typical cell phone carries 25,000 germs per inch. A kitchen sponge has 60 times the bacteria compared to a pet's food bowl. Other germ hotspots in your house include doorknobs, light switches, sink faucets, and remote controls.

So what can you do to keep these areas clean? Use a disinfectant spray or wipe on these common home surfaces once a week. If you don't have disinfecting products, you can make your own. Mix equal parts vinegar and water in a spray bottle or bucket, or combine 1 teaspoon bleach to a quart of water. Both solutions are economical and easy to make.

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• Checkpoint Solution

There are 24 hours in one full day. The formula for this exponential growth is

So, the number of bacteria in the culture after one full day is A = 46,267.

Because you cannot have a part of a bacterium, the decimal part of the number of bacteria is left off and the number of bacteria is given as a whole number.

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| \\_/ || | \\_/ || |  --  \\  | \_/ || |  /\  ||
\____//   \____//  |______//   \___//  |_// \_||
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