Question

Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption. An individual consumes two cups of coffee, each containing \(90 \mathrm{mg}\) of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours. a. Determine the amount of caffeine in the bloodstream 1 hour after drinking the first cup of coffee. b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

Short answer

Answer: a) 1 hour after drinking the first cup of coffee, there are about 68.12 mg of caffeine in the bloodstream. b) 1 hour after drinking the second cup of coffee, there are about 120.48 mg of caffeine in the bloodstream.

Step-by-step solution

Understand the exponential decay function

An exponential decay function models the amount of a substance over time given its half-life. The general form of an exponential decay function is given by \(f(t) = A \cdot 2^{-\frac{t}{h}}\), where \(A\) is the initial amount of the substance, \(t\) is the time elapsed, and \(h\) is the half-life of the substance.

Identify the given information

We are given that each cup of coffee contains 90 mg of caffeine, the half-life of caffeine for this individual is 5.7 hours, and the individual drinks two cups of coffee two hours apart. We will use the exponential decay formula to determine the amount of caffeine in the bloodstream.

Calculate caffeine amount 1 hour after the first cup

Using the given information, we have: \(A = 90 \mathrm{mg}, h = 5.7 \mathrm{h} \, \text{and} \, t = 1 \mathrm{h}\). To find the amount of caffeine in the bloodstream 1 hour after drinking the first cup, plug these values into the exponential decay formula: \(f(t) = 90 \cdot 2^{-\frac{1}{5.7}}\). Solving for \(f(1)\), we get \(f(1) \approx 68.12 \mathrm{mg}\).

Calculate caffeine amount 1 hour after the second cup

The individual consumes the second cup of coffee 2 hours after the first one. At this moment, the combined half-life of the caffeine is still 5.7 hours, but the amount of caffeine from the first cup has decreased. To find the amount of caffeine left from the first cup after 2 hours, we calculate \(f(2) = 90 \cdot 2^{-\frac{2}{5.7}} \approx 52.36 \mathrm{mg}\). Now 1 hour after drinking the second cup (3 hours after the first), we need to calculate the amount of caffeine from both cups. For the second cup, 1 hour has passed since consumption, so we calculate \(f(1)\), as in step 3: \(f(1) \approx 68.12 \mathrm{mg}\). Finally, to find the total amount of caffeine at this time, we add the amount of caffeine left from the first cup and the amount from the second cup: \(52.36 \mathrm{mg} + 68.12 \mathrm{mg} \approx 120.48 \mathrm{mg}\). #Answer# a) 1 hour after drinking the first cup of coffee, there are about 68.12 mg of caffeine in the bloodstream. b) 1 hour after drinking the second cup of coffee, there are about 120.48 mg of caffeine in the bloodstream.

Key concepts

Half-life

The concept of half-life is crucial when discussing the degradation of substances in various fields, especially radioactive decay in physics and the metabolism of drugs in pharmacology. In the context of caffeine metabolism, half-life refers to the time required for the concentration of caffeine in the bloodstream to reduce to half its initial value.

Understanding half-life is vital for determining how long a drug will affect the body, as well as planning dosing schedules to maintain or avoid certain concentration levels in the blood. The half-life can be influenced by various physiological factors, including metabolism rate, age, and body weight, which makes it an individualized measure. A shorter half-life implies that the substance will be cleared from the system more quickly, whereas a longer half-life indicates a slower elimination process.

Mathematically, the half-life is represented in the exponential decay formula as the variable 'h.' When you know the half-life and the initial quantity of the substance, you can predict the remaining quantity at any future time using this formula: \[ f(t) = A \cdot 2^{-\frac{t}{h}} \].

Calculus in Pharmacokinetics

Calculus plays a pivotal role in pharmacokinetics, the branch of pharmacology dedicated to the analysis of drug movement throughout the body. When modeling the metabolism of substances like caffeine, differential equations—specifically, exponential decay models—are employed to represent how drug concentrations in the bloodstream diminish over time.

With calculus, it is possible to calculate not just how much of a drug remains at a certain point but also the rate at which the drug's concentration is changing, which is critical for understanding peak drug efficacy and potential toxicity. Calculus allows us to determine integral values like the area under a concentration-time curve (AUC), which provides insights into the overall exposure of an organism to the given drug.

Moreover, calculus enables the optimization of dosage regimens to ensure that drug levels remain within therapeutic windows, which are the concentration ranges within which the drug exerts its desired effect without causing significant side effects.

Modeling Substance Concentration

Modeling the concentration of a substance such as caffeine in the bloodstream is an important application of calculus in the life sciences. The model not only informs us about the current amount but also predicts future concentrations and helps in understanding how the substance interacts with the body over time.

An exponential decay function is often used to model the rate at which a drug loses its potency or is cleared from the bloodstream. This model is characterized by a consistent percentage decrease over equal time increments, hence the term 'exponential.' The typical formula for calculating the remaining concentration at a point in time is\[ f(t) = A \cdot 2^{-\frac{t}{h}} \],where \(A\) represents the initial concentration, \(t\) is the time since the substance was introduced, and \(h\) is the half-life.

This model is particularly applicable to scenarios where drug absorption has ceased, and elimination is the primary phase, as with the caffeine example provided. In practice, these models assist in crafting guidelines for safe and effective medication usage, factoring in individual differences in drug metabolism.