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Thread: Exponential rate question

  1. #1

    Exponential rate question

    "A sick person takes a tablet of mass 500mg whose contents are gradually absorbed into the body. The mass not yet absorbed after t minutes is modelled by the formula 500(e^(-ct)) mg. After 30 minutes only 50mg remains unabsorbed.

    a) Find the value of c
    b) find the rate at which the contents of the tablet are being absorbed
    i) when the person first takes the tablet,
    ii) when half of the tablet is absorbed."

    I've managed to find c but unsure how to go about the rest of the question. Also, I'm unsure when questions want me to differentiate to find rates as opposed to using the idea that "A quantity Q growing (or decaying) exponentially according the the law Q= ae^(ct) has a rate of growth (or decay equal to c (or -c) times its current value". Thank you for your time, any help will be greatly appreciated.

    Edit: I found the derivative as f'(x)=-500c(e^(-ct)), and subbed in zero as x for b) i), but Im an order of magnitude too low at 4.9mg instead of 49.9mg.
    Edit: I can find the right values but an order of magnitude too low.
    Last edited by MathsHelpPlz; 12-13-2012 at 03:52 PM.

  2. #2
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    Quote Originally Posted by MathsHelpPlz View Post
    "A sick person takes a tablet of mass 500mg whose contents are gradually absorbed into the body. The mass not yet absorbed after t minutes is modelled by the formula 500(e^(-ct)) mg. After 30 minutes only 50mg remains unabsorbed.

    a) Find the value of c
    b) find the rate at which the contents of the tablet are being absorbed
    i) when the person first takes the tablet,
    ii) when half of the tablet is absorbed."

    I've managed to find c but unsure how to go about the rest of the question. Also, I'm unsure when questions want me to differentiate to find rates as opposed to using the idea that "A quantity Q growing (or decaying) exponentially according the the law Q= ae^(ct) has a rate of growth (or decay equal to c (or -c) times its current value". Thank you for your time, any help will be greatly appreciated.
    [tex]Q = ae^{ct} \implies \dfrac{dQ}{dt} = ae^{ct} * c = cQ.[/tex]

    The rule is a result of differentiation.

  3. #3
    Quote Originally Posted by JeffM View Post
    [tex]Q = ae^{ct} \implies \dfrac{dQ}{dt} = ae^{ct} * c = cQ.[/tex]

    The rule is a result of differentiation.
    Ah ok, so would the rate be 'c times the current value' rather than c?

  4. #4
    Thanks for your explanation of the rule. I understand it now and have got the correct order of magnitude.

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    Quote Originally Posted by MathsHelpPlz View Post
    Ah ok, so would the rate be 'c times the current value' rather than c?
    It is a rate so your answer will [tex]cQ\ mg\ per\ minute[/tex].

    c itself is just a dimensionless constant.

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    Quote Originally Posted by JeffM View Post
    It is a rate so your answer will [tex]cQ\ mg\ per\ minute[/tex].

    c itself is just a dimensionless constant.
    Actually, 'ct' is dimensionless - because it is used as ect.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  7. #7
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    Quote Originally Posted by Subhotosh Khan View Post
    Actually, 'ct' is dimensionless - because it is used as ect.
    Right you are. Do I have to go to the corner?

  8. #8
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    Quote Originally Posted by JeffM View Post
    Right you are. Do I have to go to the corner?
    You are already in the corner - who let you out??!!

    http://www.freemathhelp.com/forum/th...541#post324541
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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