Ron Larson
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7.2 Exponential Patterns

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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    __    __    _____     _____     _____      _____  
\ \\ / //  |  ___||  /  ___||  |  ___||   / ___// 
 \ \/ //   | ||__   | // __    | ||__     \___ \\ 
  \  //    | ||__   | \\_\ ||  | ||__     /    // 
   \//     |_____||  \____//   |_____||  /____//  
    `      `-----`    `---`    `-----`  `-----`
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    system user
    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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               _    _    _____      ______   _    _   
  ____    | || | || |  __ \\   /_   _// | \  / || 
 |    \\  | || | || | |  \ ||   -| ||-  |  \/  || 
 | [] ||  | \\_/ || | |__/ ||   _| ||_  | .  . || 
 |  __//   \____//  |_____//   /_____// |_|\/|_|| 
 |_|`-`     `---`    -----`    `-----`  `-`  `-`  
 `-`
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  • 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.)

    data folder

    • 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.

      data folder

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