A cup of coffee contains 25mg of caffeine

Question

Every hour approximately 3% of the amount of caffeine in the body is metabolized and eliminated
f(x) =

Answer 1

To model the caffeine concentration in the body over time after consuming a cup of coffee, we can set up a function that reflects the elimination rate.

Let's denote:

• \( C_0 \) as the initial amount of caffeine in the body, which is 25 mg.
• \( t \) as the time in hours since the coffee was consumed.
• \( C(t) \) as the amount of caffeine remaining in the body after \( t \) hours.

Since approximately 3% of the caffeine is metabolized every hour, 97% remains after each hour. This can be expressed as:

\[ C(t) = C_0 \cdot (0.97)^t \]

Substituting in the initial amount of caffeine:

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

This function \( C(t) \) gives you the amount of caffeine in the body in milligrams at time \( t \) hours after consuming the coffee.

Answer 2

Is f(x) the same as C(t) cause it’s asking for f(x)

Answer 3

Yes, \( f(x) \) can be used interchangeably with \( C(t) \) to represent the function that models the caffeine concentration over time.

If you prefer to use \( f(x) \) instead of \( C(t) \), you could define the function as follows:

\[ f(x) = 25 \cdot (0.97)^x \]

Here, \( x \) represents the time in hours since the coffee was consumed, and \( f(x) \) gives the amount of caffeine left in the body in milligrams after \( x \) hours.

So, the final answer can be stated as:

\[ f(x) = 25 \cdot (0.97)^x \]

This equation effectively captures the same information as the previous function \( C(t) \).