Problem 11 - Applied Calculus

Question

A cup of coffee contains about 100 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

(a) \(\frac{dA}{dt} = -0.17A\) (b) Initial rate: \(-17\, \text{mg/hr}\); change in first hour: \(-17\, \text{mg}\).

Step 1: Understanding the Problem

We are given a situation where caffeine leaves the body continuously at a rate of 17% every hour, starting after a cup of coffee containing 100 mg of caffeine is consumed. We need to model this situation with a differential equation.

Step 2: Formulate the Differential Equation

The rate of change of caffeine in the body, \(\frac{dA}{dt}\), is proportional to the amount of caffeine, \(A\), in the body at time \(t\). Since 17% of the caffeine is metabolized per hour, the differential equation is \(\frac{dA}{dt} = -0.17A\). The negative sign indicates that the caffeine amount decreases over time.

Step 3: Initial Condition

At \(t = 0\), the initial amount of caffeine \(A(0)\) is 100 mg, because that is when the coffee is consumed. Hence, \(A(0) = 100\).

Step 4: Solving the Differential Equation

For a differential equation of the form \(\frac{dA}{dt} = -0.17A\), the solution is of the exponential form \(A(t) = A_0 e^{-kt}\), where \(k = 0.17\). Thus, \(A(t) = 100e^{-0.17t}\).

Step 5: Compute the Initial Rate of Change

The initial rate of change of caffeine right after the coffee is consumed (at \(t = 0\)) can be found using \(\frac{dA}{dt} = -0.17A\). Substituting \(A = 100\) mg, we find \(\frac{dA}{dt} = -0.17 \times 100 = -17 \text{ mg/hour}\).

Step 6: Estimate Caffeine Change During the First Hour

To estimate the change in caffeine during the first hour, use the initial rate of change found in Step 5. Since \(\frac{dA}{dt} = -17\), approximately 17 mg of caffeine leaves the body during the first hour.

Key Concepts

Caffeine Metabolism

When we drink a cup of coffee, the caffeine it contains begins to impact our body. Caffeine metabolism is the process by which our body breaks down and removes caffeine. Typically, this process occurs in the liver and involves certain enzymes. This breakdown doesn't happen instantly. Instead, caffeine is metabolized over time, which means there is always a certain amount of caffeine remaining in the bloodstream after consumption.

• The metabolism of caffeine is generally slow, with about 17% of the caffeine being processed every hour.
• This percentage indicates the fraction of caffeine metabolism per unit time, making caffeine levels decrease continuously.
• It's important to model how caffeine levels reduce over time accurately, and for this, differential equations come into play.

This continuous decrease follows an exponential pattern because a percentage of the current amount, rather than a fixed amount, is metabolized every hour. This changing rate of decrease is exactly why differential equations are a helpful tool in understanding caffeine metabolism.

Exponential Decay

Exponential decay is a mathematical concept often used to describe the process through which a quantity reduces over time. In the context of caffeine metabolism, this involves modeling the percentage of caffeine that exits the body every hour.

• In a differential equation, this is represented as \(\frac{dA}{dt} = -0.17A\).
• Here, \(A\) symbolizes the amount of caffeine at any given time, and \(0.17\) represents the rate of metabolism, expressed as a decimal.
• The negative sign indicates that the caffeine amount is diminishing.

The solution to this type of differential equation is an exponential function: \(A(t) = A_0 e^{-0.17t} \). The initial condition (the caffeine amount at time zero) determines \(A_0\), or the starting point of the caffeine level. In our exercise, since the caffeine was first 100 mg, \(A_0\) equals 100.
This formula tells us how much caffeine remains at any future time, providing a comprehensive picture of caffeine's decline within the body.

Initial Conditions

Initial conditions serve as a starting point in understanding how a differential equation will unfold over time. They tell us the state of a system—in this case, the amount of caffeine—at the very beginning of observation.

• For our differential equation \(\frac{dA}{dt} = -0.17A\), it is crucial to know the initial condition: \(A(0) = 100\) mg, as given in the problem.
• This represents the amount of caffeine at the moment the coffee was consumed.

Acknowledging this initial amount is essential, as it allows us to solve the differential equation and gives meaning to how the equation behaves.
The initial condition works as a key input for calculating how the caffeine concentration decreases over time. Because of this initial condition, the solution of the differential equation ends up shaping into the form \(A(t) = 100e^{-0.17t}\).
Therefore, initial conditions don't just kickstart the equation; they are pivotal for forecasting caffeine metabolism accurately.