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Math Help - another question

  1. #1
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    another question

    A person drinks a cup of coffee. We assume tht the caffeine enters his bloodstream immediately, and that there was no caffeine in his bloodstream prior to drinking the coffee. The half-life of caffeine in a person's bloodstream is about 6 hours. A cup of coffee contains about 100 milligrams of caffeine.

    a. Let f(t) represent the number of milligrams of caffeine in the person's bloodstream at t hours after drinking the cup of coffee. Find an equation of f

    b. The person drinks the cup of coffee at 8 am and goes to bed at 11 pm Use f to predict the amount of caffeine in his bloodstream when he goes to bed.

    c. Suppose the person drinks another cup of coffee 24 hours after the first cup. How much caffeine will be in his bloodstream from these 2 cups of coffee just after drinking the second cup? Explain how you can find this result without using an equation?

    d. Now assume that the person drinks the cup of coffee at 8am and then drinks a cup of coffee every morning at 8am from then on. Sketch a qualitative graph that describes the relationship between caffeine in the person's bloodstream and time. Describe any assumptions that you make.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by getnaphd View Post
    A person drinks a cup of coffee. We assume tht the caffeine enters his bloodstream immediately, and that there was no caffeine in his bloodstream prior to drinking the coffee. The half-life of caffeine in a person's bloodstream is about 6 hours. A cup of coffee contains about 100 milligrams of caffeine.

    a. Let f(t) represent the number of milligrams of caffeine in the person's bloodstream at t hours after drinking the cup of coffee. Find an equation of f
    i don't think you've learned about exponential decay as yet, neither do i think you would want to use it anyway. so i would say:

    f(t) = A_0 2^{-t/6}

    where A_0 is the initial amount and t is the time


    b. The person drinks the cup of coffee at 8 am and goes to bed at 11 pm Use f to predict the amount of caffeine in his bloodstream when he goes to bed.
    use the equation above and plug in the corresponding value for t


    c. Suppose the person drinks another cup of coffee 24 hours after the first cup. How much caffeine will be in his bloodstream from these 2 cups of coffee just after drinking the second cup? Explain how you can find this result without using an equation?
    1 cup has 100 mg of caffeine. 24 hours is 4 periods of 6 hours each. since 6 hours is the half-life and we don't want to use the original equation, we can find how much caffeine is in the guys body from the first cup of coffee after 24 hours by:

    100 * (1/2) * (1/2) * (1/2) * (1/2) = 6.25 mg

    so after drinking a second cup, he will have 106.25 mg of caffeine in his blood stream
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  3. #3
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    thanks once again.

    You lost me with this formula. I have never seen it before. When would I use this formula? f(t)=Ao2^-t/6
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by getnaphd View Post
    thanks once again.

    You lost me with this formula. I have never seen it before. When would I use this formula? f(t)=Ao2^-t/6
    i just made it up actually.

    1/2 is 2^-1

    the half life is 6 hours, so after every 6 hours, we multiply the original amount by 1/2

    so if Ao is the original amount, after 6 hours we have Ao * 2^-1, after another 6 hours we have Ao * 2^-1 * 2^-1 and so on. so with each passing period, we multiply by 2^-1

    so i simply wanted 2^-1 to have a variable since how many times i multiply by this depends on how many periods i pass.

    so i chose t/6 as the variable so we get A_0 \left( 2^{-1} \right)^{t/6} = A_0 2^{-t/6}

    why did i choose t/6? because i want when t = 6 we multiply by 1/2

    so when t = 6, 2^{-t/6} = 2^{-6/6} = 2^-1
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