To calculate the amount of caffeine remaining in the body after a certain period, you can use the formula for exponential decay, taking into account the rate of metabolism (in this case, 3% per hour).
The formula for the remaining amount of caffeine after \( t \) hours can be expressed as:
\[ C(t) = C_0 \times (1 - r)^t \]
Where:
- \( C(t) \) is the amount of caffeine remaining after \( t \) hours.
- \( C_0 \) is the initial amount of caffeine (25 mg in this case).
- \( r \) is the decay rate per hour (3%, or 0.03 as a decimal).
- \( t \) is the time in hours.
Example Calculations
-
After 1 hour: \[ C(1) = 25 \times (1 - 0.03)^1 = 25 \times 0.97 = 24.25 \text{ mg} \]
-
After 2 hours: \[ C(2) = 25 \times (1 - 0.03)^2 = 25 \times 0.97^2 \approx 25 \times 0.9409 \approx 23.52 \text{ mg} \]
-
After 3 hours: \[ C(3) = 25 \times (1 - 0.03)^3 = 25 \times 0.97^3 \approx 25 \times 0.9127 \approx 22.82 \text{ mg} \]
-
After 4 hours: \[ C(4) = 25 \times (1 - 0.03)^4 = 25 \times 0.97^4 \approx 25 \times 0.8858 \approx 22.14 \text{ mg} \]
You can continue this process for any number of hours to find the remaining caffeine in the body. Adjust the time \( t \) as needed, and you can track how the caffeine concentration decreases over time!