# Difference equation modelling (check my work !)

1. Apr 24, 2012

### sid9221

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:

$$N_{t+1}=(N_t - \frac{N_t}{100})+0.02$$

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

$$N^* = 2$$

So the solution is:

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

Solving for Nicotine level of 1 mg

$$(\frac{99}{100})^t = \frac{1}{2}$$

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.