Question

A certain prescription drug diminishes in the system at a rate of 25% per hour. If a person was administered 1450mg of the drug, how much will remain in 4 hours? How many hours will it take for the amount of the drug in their system to be less than 5mg?

Answer 1

9514 1404 393

Answer:

• 459 mg
• about 20 hours

Explanation:

The decay factor is 1 -25% = 0.75 per hour, so the exponential equation can be written ...

r(t) = 1450·0.75^t . . . . . milligrams remaining after t hours

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a) After 4 hours, the amount remaining is ...

r(4) = 1450·0.75^4 ≈ 458.79 . . . mg

About 459 mg will remain after 4 hours.

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b) To find the time it takes before the amount remaining is less than 5 mg, we need to solve ...

r(t) < 5

1450·0.75^t < 5 . . . . use the function definition

0.75^t < 5/1450 . . . . divide by 1450

t·log(0.75) < log(1/290) . . . . . take logarithms (reduce fraction)

t > log(1/290)/log(0.75) . . . . . divide by the (negative) coefficient of t

t > 19.708

It will take about 20 hours for the amount of the drug remaining to be less than 5 mg.