To model the amount of caffeine in the body over time after consuming a cup of tea, we can represent this with a function that accounts for the initial amount of caffeine and the rate of metabolism.
Let's denote:
- the initial amount of caffeine in the body after drinking a cup of tea as \( C_0 = 25 \) mg.
- the amount of caffeine remaining in the body after \( t \) hours as \( C(t) \).
Given that approximately 3% of the caffeine is metabolized and eliminated every hour, this means that 97% of the caffeine remains after each hour. Therefore, the remaining caffeine can be expressed as an exponential decay function:
\[ C(t) = C_0 \cdot (0.97)^t \]
Substituting \( C_0 = 25 \) mg, we get:
\[ C(t) = 25 \cdot (0.97)^t \]
This function, \( f(t) = 25 \cdot (0.97)^t \), describes the amount of caffeine remaining in the body \( t \) hours after consuming the tea.