Infinite geometric series application (long)

[sjnt]

Homework Statement

Assume that the drug administered intravenously so the concentration of drug in the bloodstream jumps almost immediately to its highest level. The concentration of the drug decays exponentially.

A doctor prescribes a 240 milligram (mg), pain-reducing drug to a patient who has chronic pain. The medical instructions read that this drug should be taken every 4 hours. After 4 hours, 60% of the original dose leaves the body. Under these conditions, the amount of drug remaining in the body, at 4-hour intervals, forms a geometric series.

Homework Equations

1. Supposing that the patient takes just one dose of the medicine write an equation for the amount of the drug in the patient's blood stream t hours after taking the medicine2. How many mgs of the drug are present in the body after 4 hours? (just after second dose?)3. Graph the amount of medicine in the blood stream for the first 24 hour period.4. Show that the amount of medicine in the patient's bloodstream after the Nth dose can be expressed by a geometric series. Use sigma notation to express the series.

The Attempt at a Solution

1. Qe^-(ct) where c is a positive constant?

 

[Sethric]

Try writing out the amount of the drug in the body at each time. At time t₀, there will be 240 mg. At t=4 there will be 240 mg (next dose) + .4*240 mg (last dose). At t=8, 240 mg (next dose) + .4*240 mg (previous dose) + .4*.4*240 mg (first dose). What does the pattern look like? Can you write an equation for that?

 

[sjnt]

i modified the problem statement a bit.
I had that equation before but it has to be exponential and it has to be decaying. any thoughts?

 

[Sethric]

I see, you are talking about part 1. Well then, use your initial values. You know that the form will be y = y₀ e^-ct. You know that y₀ = 240. Next, y₄ = .4*240 = 96. That can solve for c.

 

[sjnt]

1. y=y0e^(-ct)

2. after solving for c, y=y0(2/5)^(t/4)
y(t)=240(2/5)^(4/4)+240mg=336, correct?

4. n-1
240(sigma)(2/5)^c
c=0