Another tough calc question; help

Question

The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged 13 micrograms/mililiter (mcg/mL) of the drug in their blood plasma 1 hr after a 1000-mg dosage and 4 micrograms/milliliter 5 r after dosage.

a) Find the value k, and write an equation for an exponential function that can be used to predict the concentration of the drug, in m icrograms/milliliter, t hours after a 1000-mg dosage.

b) Estimate the concentration of the drug 2 hr after a 1000-mg dosage

c_ To relieve a fever, the concentration of the rug should go no lower than 4 mcg/mL. After how many hours will a 1000-mg dosage drop to that level?

Answer 1

Solution is of the form Ce^-kt. (The problem statement doesn't say where k shows up in the formula, so I've made a guess.)

We're told that Ce^-k = 13 (since the concentration after 1 hr is 13 mcg/mL) and Ce^-5k = 4.

Therefore e^-k = 13/C; taking logarithms of both sides we get (*) -k = ln(13) - ln(C)

Similarly e^-5k = 4/C; taking logarithms of both sides we get (**) -5k = ln(4) - ln(C)

Now subtract equation (**) from equation (*). Result:

4k = ln(13) - ln(4) = ln(13/4)

Answer to (a) Therefore k = ln(13/4)/4 ≈ 0.295

C ≈ 13*e^-0.295 ≈ 17.461

(b) Concentration after 2 hrs ≈ 17.461e^-0.295*2 ≈ 9.679 mcg/mL

(c) Answer is 5 hrs. We're already told that 4 mcg/mL is the concentration after 5 hrs. The function 17.461e^-0.295*t is decreasing, so we don't reach that concentration before the 5-hr mark.