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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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  • 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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