Problem 13 - Applied Calculus

Question

A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of \(17 \%\) per hour. (a) Write a formula for the amount, \(A\) mg, of caffeine in the body \(t\) hours after drinking a cup of coffee. (b) Graph the function from part (a). Use the graph to estimate the half-life of caffeine. (c) Use logarithms to find the half-life of caffeine.

Step-by-Step Solution

Answer

The half-life of caffeine is approximately 3.73 hours.

Step 1: Understanding the Caffeine Decay Model

The amount of caffeine decreases exponentially. The formula for exponential decay is \( A(t) = A_0 \cdot e^{-kt} \), where \( A_0 \) is the initial amount of caffeine, and \( k \) is the decay constant. Here, \( A_0 = 100 \) mg, and \( 17\% \) is the decay rate per hour.

Step 2: Deriving the Decay Constant

The decay rate is \( 17\% \) which can be expressed as a decimal \( 0.17 \). Since we know this is the rate that leaves the body, we equate \( e^{-k} = 1 - 0.17 = 0.83 \). Solving for \( k \), we have: \(-k = \ln(0.83) \rightarrow k = -\ln(0.83) \approx 0.186 \).

Step 3: Writing the Formula for Caffeine Remaining

Substitute \( A_0 = 100 \) and \( k \approx 0.186 \) into the exponential decay formula: \[ A(t) = 100 \cdot e^{-0.186t} \]. This is the formula for the amount of caffeine, \( A \), in milligrams, after \( t \) hours.

Step 4: Graphing the Function

Use a graphing tool to plot the function \( A(t) = 100 \cdot e^{-0.186t} \) over a suitable range of \( t \) values, such as \( 0 \) to \( 10 \) hours. Look for the point where \( A(t) \) is approximately half the initial amount, i.e., \( 50 \) mg.

Step 5: Estimating Half-Life from the Graph

Visually inspect the graph to find the time \( t \) when \( A(t) \approx 50 \). Roughly, the graph might show that \( t \) is close to \( 4 \) or \( 5 \) hours, indicating the half-life.

Step 6: Calculating the Half-Life Algebraically

Set \( A(t) = 50 \) in the decay equation: \( 50 = 100 \cdot e^{-0.186t} \). Divide by 100: \( 0.5 = e^{-0.186t} \). Take the natural logarithm of both sides: \( \ln(0.5) = -0.186t \). Solve for \( t \) to find the half-life: \( t = \frac{\ln(0.5)}{-0.186} \approx 3.73 \).

Step 7: Verification and Conclusion

Re-check calculations to ensure accuracy. The half-life of caffeine in the body, calculated using logarithms, is approximately \( 3.73 \) hours.

Key Concepts

Caffeine Metabolism

Caffeine metabolism describes how caffeine, found in drinks like coffee, is processed by your body. When you consume caffeine, it is quickly absorbed into your bloodstream and starts its journey through your system. Caffeine affects your central nervous system, enhancing alertness and counteracting fatigue, which is why many people enjoy a morning cup of coffee.
However, your body doesn't retain caffeine indefinitely. Enzymes in your liver break down caffeine into smaller compounds, which are then excreted through urine. This process is why caffeine's effects are temporary. The rate at which caffeine is metabolized can vary based on numerous factors, such as age, liver function, and genetic makeup. Some people may clear caffeine from their systems faster than others due to these variations, affecting how long they remain alert after consumption.

Exponential Decay Formula

The exponential decay formula is crucial to understanding how substances decrease over time. For caffeine, the amount decreases at a continuous rate after ingestion. The formula for exponential decay is given by: \[ A(t) = A_0 \cdot e^{-kt} \]Where:

• \( A(t) \) is the remaining amount at time \( t \) hours.
• \( A_0 \) is the initial amount, here 100 mg.
• \( k \) is the decay constant, indicating how swiftly the substance diminishes.

In the context of caffeine, the decay constant \( k \) was found to be approximately 0.186 per hour. This constant indicates that each hour, 17% of the caffeine is metabolized by the body and only 83% remains. This formula helps us predict how much caffeine remains in our bodies as time passes, aiding in understanding its effects and duration.
Exponential decay is a fundamental concept not only in chemistry but also in biology, physics, and numerous other fields, wherever processes of decline happen at a rate proportional to their current value.

Half-Life Calculation

Half-life is a critical concept in the decay process, measuring how long it takes for half of a substance to be metabolized or decomposed. For caffeine, this involves finding the time it takes for half of the caffeine initially present in your system to be broken down. Understanding half-life helps us estimate how long caffeine's effects can be discerned within the body.
To calculate the half-life, we use the formula:\[ t = \frac{\ln(0.5)}{-k} \]In our caffeine scenario:

• \( k \) is the decay constant \( 0.186 \).
• \( \ln(0.5) \) is the natural log of 0.5, reflecting a 50% reduction.

By substituting these values into the equation, the half-life \( t \) is approximately \( 3.73 \) hours. This means after about 3.73 hours, only 50 mg out of the initial 100 mg of caffeine remains.
Grasping the concept of half-life isn't only essential for understanding caffeine metabolism but also plays a pivotal role in nuclear physics, pharmacology, and environmental science, wherever substances diminish predictably over time.