Question

A typical cup of coffee contains about 100 milligrams (mg) of
caffeine.
Every hour, approximately 6% of the amount of caffeine in the
body is metabolized and eliminated. let C represent the amount of caffeine in the body, in mg, and t
represents the number of hours since a cup of coffee was consumed. Write C as
a function of t.
C (t) =

Answer 1

To model the amount of caffeine \( C(t) \) in the body over time after consuming a cup of coffee, we need to account for the initial amount of caffeine and the rate at which it is metabolized.

1. **Initial Amount of Caffeine**: A typical cup of coffee contains about 100 mg of caffeine, so we start with:
\[
C(0) = 100 \, \text{mg}
\]

2. **Caffeine Metabolism**: Every hour, 6% of the caffeine in the body is metabolized and eliminated. This means that 94% of the caffeine remains after each hour. Therefore, the amount of caffeine remaining at any time \( t \) can be expressed as:
\[
C(t) = C(0) \times (0.94)^t
\]

3. **Putting it all together**: Since \( C(0) = 100 \) mg, we substitute that into the equation:
\[
C(t) = 100 \times (0.94)^t
\]

Thus, the function for the amount of caffeine in the body as a function of time \( t \) in hours is given by:
\[
\boxed{C(t) = 100 \times (0.94)^t}
\]