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Math Help - Applications of Calculus

  1. #1
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    Applications of Calculus

    I'm not quite sure how to approach this question, could someone help me out here? I'm sure I'll have further questions after this, so check back in this thread for updates. Anyways, the question asks;

    A certaing brand of medicine tablet is in the shape of a sphere with diameter 5mm. Ther rate at which the pill dissolves is dr/dt = -k, where r is the radius of the sphere at time t hours, and k is a positive constant.

    a) Show that r = 5/2 - kt

    b) If the pill dissolves completely in 12 hours, find k.
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  2. #2
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    i)
    \frac{dr}{dt} = -k
    r = \int -k \,dt
    r = -kt + C
    When t = 0, r = 5/2 (initially the tablet has diameter 5, which means radius is 5/2)
    \frac{5}{2} = -k(0) +C
    C = \frac{5}{2}
    Therefore r = \frac{5}{2} -kt

    ii)
    The pill is dissolved when r = 0, t = 12. Sub these into the equation we derived:

    0 = \frac{5}{2} -12k
    12k = \frac{5}{2}
    k = \frac{5}{24}
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  3. #3
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    Thanks, for some reason my mind just didn't make the leap from diameter to radius.
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  4. #4
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    Got another one;

    Twenty-five wallabies are released on Wombat Island and the population is observed over the next six years. It is found that the rate of increase in the wallaby population is given by \frac{dP}{dt} = 12t - 3t^2, and the population itself is given by P = 25 + 6t^2 - t^3. When does the population increase most rapidly?
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  5. #5
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    I'm pretty sure the population increase most rapidly means when
     \frac{d^2P}{dt^2}=0 , an inflexion where concavity changes from positive to negative

    Differentiating your rate of increase formula
     \frac{d^2P}{dt^2} = 12-6t
    12-6t = 0 for possible points of inflexion
    t = 2

    Checking the concavity of both sides of t =2, we find that concavity changes from positive to negative. Therefore there is an inflexion at t = 2

    The population increased most rapidly in the second year (assuming the year they released it is year 0)
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