The function $C(t)$ that models the amount of caffeine remaining in the body over time is generally represented by an exponential decay function. Caffeine is metabolized in the body at a rate that depends on its half-life, which averages about 5 hours for most people. The half-life is the time it takes for the caffeine level to reduce to half of its original amount.

Since we start with 390 mg of caffeine and know that half of it will remain in the body after 5 hours, we can write the function as follows:

$C(t) = 390 \times \left(\frac{1}{2}\right)^{\frac{t}{5}}$

where:

• $C(t)$ is the amount of caffeine (in mg) remaining in the body after $t$ hours,
• $t$ is the time in hours since the caffeine was consumed,
• $390$ mg is the initial amount of caffeine,
• $\left(\frac{1}{2}\right)^{\frac{t}{5}}$ models the decay based on the 5-hour half-life.

Explanation of the Function:

This function uses the exponential decay formula, where the base of the exponent is $\frac{1}{2}$ (since we’re considering half-life), and $t/5$ indicates the decay interval in terms of hours.

Summary:

The caffeine decay function for this scenario is: $C(t) = 390 \times \left(\frac{1}{2}\right)^{\frac{t}{5}}$

Would you like more details, or do you have any specific questions?

Here are five related questions to consider:

1. What is the amount of caffeine remaining in the body after 3 hours?
2. How much caffeine is left after 10 hours?
3. At what time will the caffeine level reach 50 mg?
4. How does changing the half-life affect the decay function?
5. What is the percentage of caffeine remaining after each hour?

Tip: Exponential decay functions like this one are widely applicable in fields such as pharmacology, where understanding drug metabolism is essential.