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Difference equation modelling (check my work !)

  1. Apr 24, 2012 #1
    A certain smoker has a daily intake of 0.02 milligrams of nicotine. It is assumed that
    1% of nicotine is disintegrated by the body per day.
    (i) Set up a diference equation for the amount of nicotine N_t after t days, starting with
    an initial level of N_0 = 0.
    (ii) Derive a closed form solution for Nt.
    (iii) If a concentration of 1 mg of nicotine is considered harmful, when does the smoker
    reach this threshold? How much higher does the concentration rise?

    This is my formation of the equation:

    [tex] N_{t+1}=(N_t - \frac{N_t}{100})+0.02 [/tex]

    So the steady state is [tex] N^*[\frac{1}{100}]=0.02 [/tex]

    [tex] N^* = 2 [/tex]

    So the solution is:

    [tex] N_t = (\frac{99}{100})^t [0-2]+2 [/tex]

    Solving for Nicotine level of 1 mg

    [tex] (\frac{99}{100})^t = \frac{1}{2} [/tex]

    t ~ 69 days.

    So basically you're ****ed in 69 day's (See what I did there :D)

    My only issue is with the formation as it's easy to get them wrong, hence a second opinion.

    Also as the steady state is 2, that's the upper limit that the nicotin level can reach which is also verified by the equation as for a value >2 you need to evaluate log(-) which is undefined.
     
  2. jcsd
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