### 7.2 Exponential Patterns

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• 7. Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. What does the population for year 0 represent? (See Example 3.)

• Worked-Out Solution

The population for the year 0 is 50. This means that 50 rabbits were introduced into the new area.

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```           __   __    _____    __   __    _____
____     \ \\/ //  /  ___||  \ \\/ //  /  ___||
|    \\    \ ` //  | // __     \ ` //  | // __
| [] ||     | ||   | \\_\ ||    | ||   | \\_\ ||
|  __//     |_||    \____//     |_||    \____//
|_|`-`      `-`'     `---`      `-`'     `---`
`-`
```
Guest   3 years ago |
where are the numbers?
0
• 8. Rate of Increase of a Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. At what rate is the rabbit population increasing? (See Example 3.)

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``` __   _    _    _     _  _      ___
| || | || | || | ||  | \| ||   / _ \\      ___
| '--' || | || | ||  |  ' ||  | / \ ||    /   ||
| .--. || | \\_/ ||  | .  ||  | \_/ ||   | [] ||
|_|| |_||  \____//   |_|\_||   \___//     \__ ||
`-`  `-`    `---`    `-` -`    `---`       -|_||
`-`
```
There are no comments.
• 9. Predicting a Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. Suppose the population growth continued for another 3 years. Predict the number of rabbits in year 8. (See Example 3.)

• Worked-Out Solution

From the data given in the graph, you can see that each year the rabbit population is increasing by 60%.

If this pattern continues for another year, then the population would increase as follows.

So, in year 8 there would be about 2150 rabbits.

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```  _____      ___     ______     _____     ______
/  ___||   / _ \\  |      \\  |  ___||  /_   _//
| // __    / //\ \\ |  --  //  | ||__    `-| |,-
| \\_\ || |  ___  |||  --  \\  | ||__      | ||
\____//  |_||  |_|||______//  |_____||    |_||
`---`   `-`   `-` `------`   `-----`     `-`'

```
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• 10. Predicting a Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. Suppose the population growth continued for another 3 years. Predict the number of rabbits in year 10. (See Example 3.)

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```  _  __    ______    ______    ______   __   __
| |/ //  /_   _//  /_____//  /_   _//  \ \\/ //
| ' //    -| ||-   `____ `    -| ||-    \   //
| . \\    _| ||_   /___//     _| ||_    / . \\
|_|\_\\  /_____//  `__ `     /_____//  /_//\_\\
`-` --`  `-----`   /_//      `-----`   `-`  --`
`-`
```
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• 11. Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. When does the rabbit population exceed 3000? (See Example 3.)

• Worked-Out Solution

From the data given in the graph, you can see that each year the rabbit population is increasing by 60%.

If this pattern continues, then the population would increase as follows.

So, in year 9 the population would exceed 3000 rabbits.

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``` ______    _    _     _____    __   __    ______
|      \\ | || | ||  / ____||  \ \\/ //  /_   _//
|  --  // | || | || / //---`'   \ ` //   `-| |,-
|  --  \\ | \\_/ || \ \\___      | ||      | ||
|______//  \____//   \_____||    |_||      |_||
`------`    `---`     `----`     `-`'      `-`'

```
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• 12. Rabbit Population

A rabbit population is introduced to a new area. The graph shows the growth of the rabbit population. When does the rabbit population exceed 6000? (See Example 3.)

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``` ______     ______   ______     ______
|      \\  /_   _// |      \\  /_   _//   ____
|  --  //   -| ||-  |  --  //   -| ||-   |    \\
|  --  \\   _| ||_  |  --  \\   _| ||_   | [] ||
|______//  /_____// |______//  /_____//  |  __//
`------`   `-----`  `------`   `-----`   |_|`-`
`-`
```
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• 13. Population Growth

A rabbit population grows exponentially over a 10-year period. The population in year 3 is 150. The population in year 4 is 204. Predict the number of rabbits in year 10. (See Example 3.)

• Worked-Out Solution

The percent change from year 3 to year 4 is

which is a 36% increase. You can use a spreadsheet to estimate the population in year 10.

Notice that you don't have to know the population in the years 0, 1, 2, and 3 to answer the question.

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``` _    _     ______    _  _    _    _     ______
| |  | ||  /_   _//  | \| || | || | ||  /_____//
| |/\| ||   -| ||-   |  ' || | || | ||  `____ `
|  /\  ||   _| ||_   | .  || | \\_/ ||  /___//
|_// \_||  /_____//  |_|\_||  \____//   `__ `
`-`   `-`  `-----`   `-` -`    `---`    /_//
`-`
```
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• 14. Disease Outbreak

The outbreak of a disease causes a rabbit population to decrease exponentially over a 6-year period. The population in year 2 is 1200. The population in year 3 is 960. Predict the number of rabbits in year 6. (See Example 4.)

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``` ______     ______    ______     ___     __   _
|      \\  /_   _//  /_____//   / _ \\  | || | ||
|  --  //   -| ||-   `____ `   / //\ \\ | '--' ||
|  --  \\   _| ||_   /___//   |  ___  ||| .--. ||
|______//  /_____//  `__ `    |_||  |_|||_|| |_||
`------`   `-----`   /_//     `-`   `-` `-`  `-`
`-`
```