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4.1 Exponential Growth

4.1 Exponential Growth
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  • 1. Rate of Gossip Spreading

    Gossip is informal conversation, usually about the personal details of other people's lives. Gossip tends to spread very quickly and can be distorted and exaggerated from person to person. You win the lottery and share the news with three family members. You want to keep it a secret until you cash in your ticket and do all the paperwork. Somehow your secret gets out, and the number of people who hear about your good fortune grows by 25% each hour.

    Write a formula that can be used to model the exponential growth of the gossip about you winning the lottery. (See Example 1.)

    • Worked-Out Solution

      The formula for this exponential growth is

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    __    __   __   __    ______    _____     ______  
    \ \\ / //  \ \\/ //  /_   _//  |  ___||  /_   _// 
     \ \/ //    \ ` //   `-| |,-   | ||__    `-| |,-  
      \  //      | ||      | ||    | ||__      | ||   
       \//       |_||      |_||    |_____||    |_||   
        `        `-`'      `-`'    `-----`     `-`'   
                                                      
    
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  • 2. Rate of Gossip Spreading

    Gossip is informal conversation, usually about the personal details of other people's lives. Gossip tends to spread very quickly and can be distorted and exaggerated from person to person. You win the lottery and share the news with three family members. You want to keep it a secret until you cash in your ticket and do all the paperwork. Somehow your secret gets out, and the number of people who hear about your good fortune grows by 25% each hour.

    Use the formula from Exercise 1 to complete the table. (See Example 1.)

    data folder

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     _    _     _____    _    _      ___      _____   
    | |  | ||  |  ___|| | |  | ||   / _ \\   /  ___|| 
    | |/\| ||  | ||__   | |/\| ||  | / \ || | // __   
    |  /\  ||  | ||__   |  /\  ||  | \_/ || | \\_\ || 
    |_// \_||  |_____|| |_// \_||   \___//   \____//  
    `-`   `-`  `-----`  `-`   `-`   `---`     `---`   
                                                      
    
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  • 3. Rate of Gossip Spreading

    Gossip is informal conversation, usually about the personal details of other people's lives. Gossip tends to spread very quickly and can be distorted and exaggerated from person to person. You win the lottery and share the news with three family members. You want to keep it a secret until you cash in your ticket and do all the paperwork. Somehow your secret gets out, and the number of people who hear about your good fortune grows by 25% each hour.

    The gossip about you winning the lottery continues to grow exponentially. How many people will know about you winning the lottery after 24 hours? (See Example 1.)

    • Worked-Out Solution

      The formula for this exponential growth is

      The number of people who have heard the gossip after 24 hours is

      You can check your work by entering the information in a spreadsheet.

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     __   _      ___      ______     ___      _____   
    | || | ||   / _ \\   /_   _//   / _ \\   /  ___|| 
    | '--' ||  / //\ \\  `-| |,-   / //\ \\ | // __   
    | .--. || |  ___  ||   | ||   |  ___  ||| \\_\ || 
    |_|| |_|| |_||  |_||   |_||   |_||  |_|| \____//  
    `-`  `-`  `-`   `-`    `-`'   `-`   `-`   `---`   
                                                      
    
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  • 4. Rate of Gossip Spreading

    Gossip is informal conversation, usually about the personal details of other people's lives. Gossip tends to spread very quickly and can be distorted and exaggerated from person to person. You win the lottery and share the news with three family members. You want to keep it a secret until you cash in your ticket and do all the paperwork. Somehow your secret gets out, and the number of people who hear about your good fortune grows by 25% each hour.

    Suppose the gossip growth rate is changed from 25% each hour to 50% each hour. Does the second column of the table double? Explain your reasoning. (See Example 1.)

    data folder

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      _  _     _____     _____    _    _     _____   
     | \| ||  |  ___||  / ____|| | || | ||  /  ___|| 
     |  ' ||  | ||__   / //---`' | || | || | // __   
     | .  ||  | ||__   \ \\___   | \\_/ || | \\_\ || 
     |_|\_||  |_____||  \_____||  \____//   \____//  
     `-` -`   `-----`    `----`    `---`     `---`   
                                                     
    
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  • 5. Celebrity Wedding Gossip

    As a result of the Internet and modern technology, gossip spreads much faster today than it did a few decades ago. This is especially true with news from Hollywood. Suppose a celebrity has a private, unannounced wedding with 100 guests. Once the news breaks, the number of people who know about the wedding grows by 45% each minute. Use a spreadsheet to illustrate the number of informed people every 3 minutes for up to 30 minutes. (See Example 1 and Example 2.)

    • Worked-Out Solution

      The formula for this exponential growth is

      The completed spreadsheet is shown below.

      data

      Some types of rumors occur as intentional hoaxes. Here are five common types of hoaxes.

      1. Celebrity deaths
      2. Tragic events
      3. Political rumors
      4. Facebook rumors
      5. Tech rumors
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     _____      _____   __    __     ___      ______  
    |  __ \\   |  ___|| \ \\ / //   / _ \\   /_____// 
    | |  \ ||  | ||__    \ \/ //   / //\ \\  `____ `  
    | |__/ ||  | ||__     \  //   |  ___  || /___//   
    |_____//   |_____||    \//    |_||  |_|| `__ `    
     -----`    `-----`      `     `-`   `-`  /_//     
                                             `-`      
    
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  • 6. Social Networking

    Do you think social networking websites and blogs are effective ways to communicate or get information? Explain your reasoning.

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