Exp Decay for Applied Calculus

Question

The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical​ study, adult subjects averaged

15 micrograms/milliliter (mcg/mL) of the drug in their blood plasma 1 hr after a​ 1000-mg dosage and 4 micrograms/milliliter 7 hr 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​ micrograms/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 drug should go no lower than 2 mcg/mL. After how many hours will a​ 1000-mg dosage drop to that​ level?

Answer 1

a) At 1 hour there is 15mcg/mL. After an hour passes we're left with only a fraction of the concentration. It looks like they're calling the fraction k. At 1 hour we have 15. So at 2 hours, we have 15*k. After another hour passes (3 hours), we take another fraction of the concentration, so we'll have (15*k)*k or 15k².

After 4 hours pass 15k³, and so on. At the seventh hour, we'll have 15k⁶. We also know that at the seventh hour, we have 4 mcg/mL. Therefore 15k⁶ = 4. Solve for k.

Your equation is y = A*k^t. y and t are the variables. y is the concentration and t is the hours. You have k now so you have to find A. You can do this by plugging in the point (1, 15) since at 1 hour the concentration was 15. Next solve for A. Last write out the equation with the A and k you solved for.

b) Plug in 2 for t in your equation and solve for y

c) Plug in 2 for y and solve for t.

Answer 2

So Ellen G.,

You simply have to be able to work with exponentials, they're ubiquitous in math and science.

If [S(t)] = serum level in mcg/ml after 1 g oral dosage = a*exp(-kt), then ln S(t) = (ln a) -kt, I think is the way it goes. Anyway, if you ratio the levels at two different times, the ratioing of the exponential terms is exp(k(t2-t1)). Then taking the ln of that levels ratio, results in k*(t2-t1) .

You should be able to take that relationship and use the given conditions to solve for k, since t1=1 hr and t2=7 hr ??

The other parts of the problem solve similarly; k is the value you find in part 1.

-- Cheers, --Mr. d