Solve an exponential word problem

Question

The biological half-life of caffeine is about 6hr. If one cup of coffee has 80 mg of caffeine, then the amount of caffeine C (in mg) remaining after t hours is given by C=80(2)^-t/6

A. How long will it take for the amount of caffeine to drop below 60 mg? Round to 1 decimal place.

B. Laura had trouble sleeping if she had more than 30 mg of caffeine in her bloodstream. How many hours after drinking a cup of coffee would Laura have to wait so that the coffee wouldn’t disrupt her sleep? Round to 1 decimal place.

Answer 1

C = 80(2)^-t/6

A. How long will it take for the amount of caffeine to drop below 60 mg? Round to 1 decimal place.

C is the amount of caffeine so we want C = 60:

60 = 80(2)^-t/6

3/4 = (2)^-t/6

log₂(3/4) = -t/6

Solve for t. Use your calculator to compute the answer. Round to one decimal place. The units are hours.

B. Laura had trouble sleeping if she had more than 30 mg of caffeine in her bloodstream. How many hours after drinking a cup of coffee would Laura have to wait so that the coffee wouldn’t disrupt her sleep? Round to 1 decimal place.

Again, we want C = 30 mg

30 = 80(2)^-t/6

Solve for t just like we did in part A. Another way to do it is to note that 30 mg equals one half of 60 mg, so it will take one half-life period (6 hours) to get to 30 mg after it gets to 60 mg.