Question

A patient is given a dosage Q mg/ml of a drug at regular intervals of time T hours. Assume that the drug enters the system immediately upon ingestion and that the decrease in the concentration in the blood over time is proportional to the concentration itself.

a) If the first dose is administered at , find the amount remaining in the blood at time T. This is known as the residual, R1 .

b) Find R2, R3, and the nth residual Rn.

Any help would be much appreciated!!

Answer 1

We need to try and break this problems down by converting each given fact into a mathematical model. Considering you're aware that this is a calculus question, the statement "DECREASE in the concentration in the blood OVER TIME is PROPORTIONAL to the concentration itself" should be a sign that we're dealing with a rate of change.

If two variables are proportional, that means one is a constant multiple of the other (and vice versa). Considering our quantity is Q and we know we're talking about a decrease over time, we should realise that our model is of the form
dQ/dt = -kQ
where Q is the quantity and k is a positive constant. We have a minus sign before the k because it's a decrease over time; if it was an increase, there would be no minus sign.

From this, we know that Q must be of the form Q = Ae^(-kt), where A is a constant that represents the initial quantity and k is the same constant from before. This is something you just have to remember (or investigate in a Differential Equations course) but you can prove it by back-checking; differentiate Q and see if you can form the original model equation.

For a) you wrote "If the first dose is administered at ," so that piece of information that's missing would be pretty important. In any case, try using the information I've given you (and a bit of algebra/problem solving) to work out the question.