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4.3 Exponential Decay

4.3 Exponential Decay
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  • 19. Social Media

    A video posted on a social media website has 1024 views on day 1. The number of views increases by 50% each day through day 10, and then decreases by 10% each day through day 30. Write a formula that represents the number of daily views for the first 10 days.

    • Worked-Out Solution

      Use the formula for exponential growth.

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      ______     ___      _____      ___     _____    
     /_____//   / _ \\   / ____||   / _ \\  |  __ \\  
     `____ `   / //\ \\ / //---`'  / //\ \\ | |  \ || 
     /___//   |  ___  ||\ \\___   |  ___  ||| |__/ || 
     `__ `    |_||  |_|| \_____|| |_||  |_|||_____//  
     /_//     `-`   `-`   `----`  `-`   `-`  -----`   
     `-`                                              
    
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  • 20. Social Media

    A video posted on a social media website has 1024 views on day 1. The number of views increases by 50% each day through day 10, and then decreases by 10% each day through day 30. How many views does the video have on day 10?

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               __   __   _    _    __   __   _    _   
        ___    \ \\/ // | || | ||  \ \\/ // | \  / || 
       /   ||   \ ` //  | || | ||   \ ` //  |  \/  || 
      | [] ||    | ||   | \\_/ ||    | ||   | .  . || 
       \__ ||    |_||    \____//     |_||   |_|\/|_|| 
        -|_||    `-`'     `---`      `-`'   `-`  `-`  
         `-`                                          
    
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  • 21. Social Media

    A video posted on a social media website has 1024 views on day 1. The number of views increases by 50% each day through day 10, and then decreases by 10% each day through day 30. Write a formula that represents the number of daily views for day 10 through day 30.

    • Worked-Out Solution

      Use the formula for exponential decay.

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     _    _     _____     _____     _____     ______  
    | || | ||  |  ___||  / ____||  |  ___||  /_   _// 
    | || | ||  | ||__   / //---`'  | ||__      | ||   
    | \\_/ ||  | ||__   \ \\___    | ||__     _| ||   
     \____//   |_____||  \_____||  |_____||  /__//    
      `---`    `-----`    `----`   `-----`   `--`     
                                                      
    
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  • 22. Social Media

    A video posted on a social media website has 1024 views on day 1. The number of views increases by 50% each day through day 10, and then decreases by 10% each day through day 30. Sketch a graph showing the number of daily views for the 30 days.

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     __   _    __   __   _    _     ______            
    | || | ||  \ \\/ // | || | ||  /_   _//     ___   
    | '--' ||   \ ` //  | || | ||   -| ||-     /   || 
    | .--. ||    | ||   | \\_/ ||   _| ||_    | [] || 
    |_|| |_||    |_||    \____//   /_____//    \__ || 
    `-`  `-`     `-`'     `---`    `-----`      -|_|| 
                                                 `-`  
    
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  • 23. Social Media

    A video posted on a social media website has 1024 views on day 1. The number of views increases by 50% each day through day 10, and then decreases by 10% each day through day 30. Does the video have more views on day 1 or day 30?

    • Worked-Out Solution

      Use the formula for exponential decay.

      Notice that the 10th day is represented by n = 0.

      By this reasoning, the 30th day is represented by n = 20.

      So, with a 10% decrease each day for days 11 through 30, you would expect the number of views to have dropped to around 4800. The video had only 1024 views on day 1. So, it had more views on day 30.

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     __   _      ___     __   _     _____     _____   
    | || | ||   / _ \\  | || | ||  |  ___||  /  ___|| 
    | '--' ||  | / \ || | '--' ||  | ||__   | // __   
    | .--. ||  | \_/ || | .--. ||  | ||__   | \\_\ || 
    |_|| |_||   \___//  |_|| |_||  |_____||  \____//  
    `-`  `-`    `---`   `-`  `-`   `-----`    `---`   
                                                      
    
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  • 24. Radioisotope Dating

    There are several other methods that use radioactive decay to date objects. When a radioactive substance decays into another substance, the original substance is called the "parent" and the new substance is called the "daughter."

    a. When is the number of parents the same as the number of daughters?

    b. After two half-lives, what is the ratio of the number of parents to the number of daughters?

    c. Does the graph of the parents exhibit exponential decay? Does the graph of the daughters exhibit exponential growth? Explain.

    d. Several methods of dating objects are shown in the diagram. Why do you think uranium-lead dating can be used to date older objects that carbon-14 dating cannot date?

     

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