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4.3 Exponential Decay

4.3 Exponential Decay
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  • 7. Saber-Toothed Cat

    The half-life of a snake's venom is the amount of time it takes for 50% of the venom to be removed from an animal's body.

    A snake bit a saber-toothed cat, injecting 20 milligrams of venom into its blood. The venom had a half-life of 40 hours. Assume the saber-toothed cat was still alive after 80 hours. How much of the venom remained in its bloodstream? (See Example 3 and Example 4.)

    • Worked-Out Solution

      Use the formula for half-life.

      After 80 hours, there is still 5 milligrams of the venom left in the cat's bloodstream.

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     _    _     ______    _  _    __   __  __    __  
    | \  / ||  /_   _//  | \| ||  \ \\/ // \ \\ / // 
    |  \/  ||   -| ||-   |  ' ||   \ ` //   \ \/ //  
    | .  . ||   _| ||_   | .  ||    | ||     \  //   
    |_|\/|_||  /_____//  |_|\_||    |_||      \//    
    `-`  `-`   `-----`   `-` -`     `-`'       `     
                                                     
    
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  • 8. Tasmanian Tiger

    The half-life of a snake's venom is the amount of time it takes for 50% of the venom to be removed from an animal's body. A snake bit a Tasmanian tiger, injecting 40 milligrams of venom into its blood. The venom had a half-life of 16 hours. Assume the Tasmanian tiger was still alive after 2 days. How much of the venom remained in its bloodstream? (See Example 3 and Example 4.)

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     _    _    __   __   _    _      ___     __   __  
    | || | ||  \ \\/ // | \  / ||   / _ \\   \ \\/ // 
    | || | ||   \ ` //  |  \/  ||  / //\ \\   \   //  
    | \\_/ ||    | ||   | .  . || |  ___  ||  / . \\  
     \____//     |_||   |_|\/|_|| |_||  |_|| /_//\_\\ 
      `---`      `-`'   `-`  `-`  `-`   `-`  `-`  --` 
                                                      
    
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  • 9. Snake Venom

    The half-life of a snake's venom is the amount of time it takes for 50% of the venom to be removed from an animal's body.

    A snake bit a saber-toothed cat, injecting 20 milligrams of venom into its blood. The venom had a half-life of 40 hours. A snake bit a Tasmanian tiger, injecting 40 milligrams of venom into its blood. The venom had a half-life of 16 hours. How long did it take the amount of venom in each animal's bloodstream to drop to 10 milligrams? ( See Example 3 and Example 4.)

    A spreadsheet is available to help you with this exercise.

    • Worked-Out Solution

      Enter the following formulas into a spreadsheet.

      Saber-toothed cat

      Tasmanian tiger

      For the saber-toothed cat, it takes about 40 hours for the amount of venom to drop to 10 milligrams.

      For the Tasmanian tiger, it takes about 32 hours for the amount of venom to drop to 10 milligrams.

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     _    _     _____    _    _     ______   _    _   
    | || | ||  |  ___|| | |  | ||  /_   _// | |  | || 
    | || | ||  | ||__   | |/\| ||   -| ||-  | |/\| || 
    | \\_/ ||  | ||__   |  /\  ||   _| ||_  |  /\  || 
     \____//   |_____|| |_// \_||  /_____// |_// \_|| 
      `---`    `-----`  `-`   `-`  `-----`  `-`   `-` 
                                                      
    
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  • 10. Carbon Dating

    The venom eventually killed the saber-toothed cat. An archaeologist discovers the remains and calculates that the ratio of carbon-14 to carbon-12 is about one-eighth of that occurring in the atmosphere. How long ago did the saber-toothed cat live? (See Example 3 and Example 4.)

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      ______   __   __    _  __   __   __    ______  
     /_   _//  \ \\/ //  | |/ //  \ \\/ //  /_   _// 
     `-| |,-    \ ` //   | ' //    \ ` //     | ||   
       | ||      | ||    | . \\     | ||     _| ||   
       |_||      |_||    |_|\_\\    |_||    /__//    
       `-`'      `-`'    `-` --`    `-`'    `--`     
                                                     
    
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  • 11. Carbon Dating

    The venom eventually killed the Tasmanian tiger. An archaeologist discovers the remains and calculates that the Tasmanian tiger lived 500 years ago. Estimate the ratio of carbon-14 to carbon-12 in the remains. (See Example 3 and Example 4.)

    • Worked-Out Solution

      For a time of 500 years, the ratio of carbon-14 to carbon-12 in the remains is given by

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     ______     _____     _____      ___     ______   
    |      \\  |  ___||  / ____||   / _ \\  |      \\ 
    |  --  //  | ||__   / //---`'  | / \ || |  --  // 
    |  --  \\  | ||__   \ \\___    | \_/ || |  --  \\ 
    |______//  |_____||  \_____||   \___//  |______// 
    `------`   `-----`    `----`    `---`   `------`  
                                                      
    
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  • 12. Snake Venom

    A snake bit a saber-toothed cat, injecting 20 milligrams of venom into its blood. Suppose the half-life of the venom injected into the saber-toothed cat was 20 hours. Would this increase or decrease the answer to Exercise 7? Explain. (See Example 3 and Example 4.)

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     ______      ___    __    __    ______   _    _   
    |      \\   / _ \\  \ \\ / //  /_   _// | |  | || 
    |  --  //  | / \ ||  \ \/ //    -| ||-  | |/\| || 
    |  --  \\  | \_/ ||   \  //     _| ||_  |  /\  || 
    |______//   \___//     \//     /_____// |_// \_|| 
    `------`    `---`       `      `-----`  `-`   `-` 
                                                      
    
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