Problem 11 - Applied Calculus

Question

A cup of coffee contains about \(100 \mathrm{mg}\) of caffeine. Caffeine is metabolized and leaves the body at a continuous rate of about \(17 \%\) every hour. (a) Write a differential equation for the amount, \(A\), of caffeine in the body as a function of the number of hours, \(t\), since the coffee was consumed. (b) Use the differential equation to find \(d A / d t\) at the start of the first hour (right after the coffee is consumed). Use your answer to estimate the change in the amount of caffeine during the first hour.

Step-by-Step Solution

Answer

The differential equation is \(\frac{dA}{dt} = -0.17A\) and the rate of change initially is \(-17\) mg/hr.

Step 1: Understand the differential equation

To model the rate of caffeine leaving the body, we set up the differential equation.The rate of change of caffeine, \(\frac{dA}{dt}\), is proportional to the amount of caffeine present. This can be expressed as:\[\frac{dA}{dt} = -kA\]where \(k\) is the rate constant.

Step 2: Determine the rate constant, k

We are told that caffeine decreases by 17% every hour. So, \(k = 0.17\). This means the differential equation becomes:\[\frac{dA}{dt} = -0.17A\]

Step 3: Apply initial conditions to find the rate of change

At time \(t = 0\), the amount of caffeine \(A(0)\) is 100 mg. Substituting into the equation gives:\[\frac{dA}{dt} = -0.17 \times 100 = -17\]

Step 4: Interpret the results

The rate of change of caffeine at the start (immediately after drinking) is \(-17\) mg/hr, meaning the caffeine level decreases by 17 mg in the first hour.

Key Concepts

Caffeine Metabolism

Caffeine metabolism is the process by which the body breaks down and eliminates caffeine, a stimulating compound found in coffee, tea, and other food items. When you drink a cup of coffee, caffeine enters your bloodstream and is eventually metabolized by your liver.
During this process, specific enzymes help change caffeine into different compounds that your body can excrete. With time, your caffeine levels decrease, which can impact how energetic or alert you feel.

• The liver metabolizes caffeine, converting it into three main metabolites: paraxanthine, theobromine, and theophylline.
• Different people metabolize caffeine at different rates, depending on genetic factors and other aspects like age, sex, or even pregnancy.

Caffeine metabolism can be modeled using differential equations to understand how quickly caffeine is removed from the body over time.

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current value. In the case of caffeine metabolism, this describes how caffeine levels decrease in your body over time.

• The formula for exponential decay is \ \[ \frac{dA}{dt} = -kA \] where \ \( A \) is the amount of substance remaining, \ \( k \) is the decay constant, and \ \( \frac{dA}{dt} \) is the rate of change of the substance over time.
• A negative sign indicates a decrease in the substance over time.

In our coffee example, we determine that \ \( k \) equals 0.17, meaning caffeine levels drop by 17% every hour. This rapid decrease is typical of exponential decay, where the rate of decline slows as the quantity gets smaller because there is less substance left to decrease.

Initial Conditions

When dealing with differential equations, initial conditions are a crucial piece. They allow us to solve these equations for specific scenarios. Initial conditions specify the value of a function at a particular point, which is essential for finding a unique solution to a differential equation.
For the caffeine problem, the initial condition is when \ \( t = 0 \), the initial amount of caffeine is 100 mg. Setting this condition helps us calculate the rate of change in caffeine right after drinking the coffee.

• Using the initial condition, we substitute into the equation: \ \[ \frac{dA}{dt} = -0.17 \times 100 = -17 \]
• This tells us that immediately after consumption, caffeine decreases at a rate of 17 mg per hour.

Initial conditions are essential in making predictions and interpreting results in real-life scenarios, such as estimating the caffeine decay during the first hour after consumption.