Question

Calculus help: (Question in reply section)
Codeine phosphate is a drug used as a painkiller. A common brand contains 30 mg of codeine. Samples of blood were taken at regular time intervals from a patient who had taken a pill containing 30 mg of codeine. The amount of codeine in the bloodstream was determined every 30 min for 3 h. The data are shown in the table below.
Time After Consumption (min): 30, 60, 90, 120, 150, 180
Amount of Codeine in Blood (mg): 27.0, 23.5, 21.2, 18.7, 16.6, 14.5

Answer 1

I figured out the answer for parts a and b, but I need some helpl with part c.

a) Create a scatter plot of the data and determine a suitable equation to model the amount of codeine in the bloodstream t min after taking the pill. Justify your choice of models.

b) Use the model to determine the instantaneous rate of change in the amount of codeine at each time given in the chart. How does it relate to the amount of codeine in the blood?

c) It is recommended that a second pill be taken when 90% of the codeine is eliminated from the body. When would this occur?

Answer 2

you would want to find when on your eqn 90% of the drug is out of the blood...
\[30-(30*0.90)=3\] so we know y=3; find where on your eqn that would be with x (time in min)

Answer 3

Oh ok so my equation is A=31.0213e^-0.00463t. So would I do 3=31.0213e^-0.00463t and solve for t? Or would I use the derivative of my equation which is A'=-0.1436e^-0.00463t and have the A' value equal 3?

Answer 4

Assume an exponential model from the start, so that the amount of codeine \(A\) at time \(t\) is given by \(A=Ce^{kt}\).

We're given a few points, which is nice, but we only need two. Take the first two, these give the system
\[\large\begin{cases}
27.0=Ce^{30k}\\
23.5=Ce^{60k}
\end{cases}~~\Rightarrow~~C=31.0213,~k=-0.00463\]
(just wanted to check your work for myself :P)

Instantaneous rate of change is just the derivative, looks like you're all over that.

The second pill should be taken when 90% of the codeine is gone, i.e. when there is 10% of the initial dosage left. The initial dosage was 30 mg (well, closer to 31.0213 according to the model, but whatever :P), and 10% of 30 is 3. This means you want to find the time \(t\) such that
\[\large 3=31.0213e^{-0.00463t}\]
which apparently happens at \(t\approx505\) minutes.