Question

A typical cup of tea contains about 25 milligrams (mg) of caffeine. Every hour, approximately 3% of the amount of caffeine in the body is metabolized and eliminated.

decay or growth
r-value?
function?

Answer 1

The scenario you've described involves the metabolism and elimination of caffeine from the body over time, which is a classic example of exponential decay rather than growth.

### Decay or Growth
- **Decay:** In this case, the amount of caffeine decreases over time as the body metabolizes it.

### r-value
The decay rate \( r \) is a key aspect of exponential decay. Since 3% of the caffeine is eliminated every hour, the r-value can be expressed in relation to this percentage.

The decay formula is typically:

\[
y(t) = y_0 e^{-rt}
\]

Where:
- \( y(t) \) is the amount of caffeine at time \( t \).
- \( y_0 \) is the initial amount of caffeine (25 mg).
- \( r \) is a positive constant representing the decay rate.
- \( t \) is the time in hours.

Given that 3% is eliminated every hour, 97% remains. This can be expressed as:

\[
y(t) = 25 \times (0.97)^t
\]

The \( r \) value can be found using natural logarithms. If 3% is eliminated, 97% remains, which gives us:

\[
r = -\ln(0.97) \approx 0.030459
\]

Where \( -\ln(0.97) \) gives a positive decay constant.

### Function
The function for the amount of caffeine \( C(t) \) in milligrams over time \( t \) (in hours) can be expressed as:

\[
C(t) = 25 \times (0.97)^t
\]

Alternatively, using the exponential function, it could be expressed as:

\[
C(t) = 25 e^{-0.030459t}
\]

Both functions describe the same exponential decay of caffeine in the body over time.