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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

3 answers

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.

f(x) =

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.