Algebra Caffeine problem

Question

Rebecca drinks a cup of coffee every morning at 5:00 am. The amount of caffeine (in mg) in her body as a result of drinking this coffee is given by the function 𝐴(𝑡) = 200(1/2)^t/6, where 𝑡 is time in hours since 5:00 am.

1. What is an appropriate domain for this function?
2. Sketch an accurate graph of this function in an appropriate window, labeling axes and scale.

c. What is the 𝑦-intercept of this function? What does this represent?

d. After how many hours will there be half of the original amount of caffeine left in her body, assuming she doesn’t ingest any more during the day?

Answer 1

The domain is the possible values of "t" (the time in hours after 5:00 am). So it seems reasonable to me that the smallest value "t" can be just after zero and, since she drinks coffee every day, that the largest value of "t" would be 24 (since at 24 hours, she would drink another cup the next day) . You can put that in interval notation as (0, 24]

Use your graphing calculator or desmos to enter the function y = 200(1/2)^t/6 and you'll get a graph that looks like this:

The y-intercept is the value of the caffeine level initially after she drinks her coffee (at time t = 0) and is 200 mg.

To find out when (the value of "t") the caffeine is half gone (or is 100 mg) you can look at the equation. 200 times 1/2 would give 100 so you want the value of "t" that make (1/2)^t/6 equal to 1/2. Can you see that is 6? (because (1/2)^6/6 = (1/2)¹ = 1/2)