Question

A patient is given a 100-milligram dosage of a drug that decays exponentially at a rate of 11% per hour. The doctor wants to know when 15 milligrams of the drug remains.

Write an equation that models the situation. Explain each aspect of your equation.

Answer 1

16.3 hours.

Explanation:

"Decays exponentially at a rate of 11% per hour" translates into

(1 - 0.11)^h, where h is the number of hours during which the drug decays.

The negative sign comes from "decay."

The complete formula applicable here is

d(h) = (100 mg)(1 - 0.11)^h, where h is the number of hours.

We want to know how many hours it will be until all but 15 mg remain. The appropriate formula for this is

15 mg = (100 mg)()^h. Dividing both sides by 100 mg, we get:

15 mg

------------- = 0.89^h, or 0.15 = 1.11^h

100 mg

Apply properties of logs to solve this. Take the natural log of both sides of the above equation. We get:

ln 0.15 = h(ln 0.89). Dividing both sides by ln 0.89, we get:

h = 16.3

The original 100 mg dosage would decay to 15 mg after 16.3 hours.