Question

A typical cup of tea contour 25 milligrams of caffeine
Every hour approximately 3% of the amount of caffeine in the body is metabolized and eliminated
Determine decay or growth
r-value
Function

Answer 1

f(x) =

Answer 2

To determine the decay of caffeine in the body after consuming a typical cup of tea, we start with the amount of caffeine ingested and the rate at which it's metabolized.

1. **Amount of caffeine**: \( C_0 = 25 \) milligrams.
2. **Decay rate**: Given that 3% of the caffeine in the body is metabolized each hour, this translates into a decay constant \( r \).

### Step 1: Define the decay rate
The decay rate can be expressed as \( r = -0.03 \) (where the negative sign indicates decay). This means that each hour, the amount of caffeine left is 97% of its previous amount.

### Step 2: Define the function
The amount of caffeine at time \( t \) (in hours) can be modeled using the exponential decay formula:

\[
C(t) = C_0 e^{rt}
\]

Substituting in our values, we have:

- \( C_0 = 25 \) mg
- \( r = -0.03 \)

Thus, the function becomes:

\[
C(t) = 25 e^{-0.03t}
\]

### Step 3: Conclusion
1. **Decay or growth**: This is a decay function due to the negative value of \( r \).
2. **r-value**: The r-value is \( -0.03 \).
3. **Function**: The function that describes the amount of caffeine left in the body after \( t \) hours is:

\[
C(t) = 25 e^{-0.03t}
\]

This function indicates that the amount of caffeine decreases over time according to the given decay rate.

Answer 3

If you are looking for a representation of the caffeine decay function as \( f(x) \), we can express it similarly to how we defined it previously, but with the variable changed from \( t \) (for time) to \( x \).

Using the same parameters we discussed earlier, the function that describes the amount of caffeine left in the body after \( x \) hours would be:

\[
f(x) = 25 e^{-0.03x}
\]

Where:
- \( f(x) \) represents the amount of caffeine (in milligrams) in the body after \( x \) hours.
- \( 25 \) is the initial caffeine amount consumed (in mg).
- \( -0.03 \) is the decay rate.
- \( e \) is Euler's number (approximately 2.71828).

This function quantifies how the caffeine level decreases over time due to metabolic processes.