Problem 9 - Applied Calculus

Question

Alcohol is metabolized and excreted from the body at a rate of about one ounce of alcohol every hour. If some alcohol is consumed, write a differential equation for the amount of alcohol, \(A\) (in ounces), remaining in the body as a function of \(t\), the number of hours since the alcohol was consumed.

Step-by-Step Solution

Answer

The differential equation is \( \frac{dA}{dt} = -1 \).

Step 1: Understand the Problem

We know that alcohol leaves the body at a constant rate of one ounce per hour. We need to express this process using a differential equation for the amount of alcohol, \(A\).

Step 2: Define Variables and Rate of Change

Let \(A(t)\) be the amount of alcohol in the body at time \(t\) in ounces. The rate of change of \(A\) with respect to time \(t\) is given as a constant rate of -1 ounce per hour, indicating the alcohol is leaving the body.

Step 3: Formulate the Differential Equation

The rate of change of \(A(t)\) is constant and negative, as alcohol is leaving the body. The differential equation representing this process can be written as: \[ \frac{dA}{dt} = -1 \]

Step 4: Check the Differential Equation

Ensure that the formulated differential equation represents the scenario described: Alcohol is leaving the body at a constant rate of 1 ounce per hour, which is accurately captured by \(\frac{dA}{dt} = -1\).

Key Concepts

Rate of Change

Understanding the concept of the rate of change is essential when dealing with differential equations, especially in the context of metabolism. In simple terms, the "rate of change" describes how quickly a quantity, such as the amount of alcohol in the body, is changing over time. For our example, the rate at which alcohol is metabolized is known to be consistent: one ounce is metabolized per hour.
This kind of problem is commonly interpreted through the language of calculus. Specifically, the rate of change of alcohol remaining in the body with respect to time is negative because the amount of alcohol is decreasing. This is represented mathematically as a differential equation:

• \( rac{dA}{dt} = -1 \) - Here, \( dA \) represents a small change in the amount of alcohol, and \( dt \) is a small change in time.
• The negative sign reflects the reduction in alcohol content over time.

To visualize this, imagine a graph where the y-axis is the alcohol amount and the x-axis is time. The slope of this graph shows how fast the alcohol is being metabolized. Given the constant rate, the graph would be a straight line sloping downward, illustrating continual decrease.

Metabolism

Metabolism plays a crucial role in how substances like alcohol are processed by the body. Metabolism in the context of alcohol refers to the biochemical process through which the body breaks down alcohol. This process is primarily carried out in the liver. For educational purposes, let’s simplify this to understand its connection with differential equations.
When alcohol enters the body, metabolism acts similarly to a machine consistently working to rid the body of alcohol. The steady pace at which this machine works is crucial. In mathematical terms, a fixed metabolic rate of one ounce per hour is expressed as constant in our differential equation. It is critical to notice:

• This rate is fixed, meaning it does not speed up or slow down.
• The consistency of this rate allows for predictable outcomes, helping us formulate the differential equation \( \frac{dA}{dt} = -1 \).

This predictable metabolic behavior simplifies the modeling of physical processes involving steady-state scenarios—ideal for students learning about differential equations and how they apply to real-world situations.

Time Function

Time functions are mathematical tools that describe how quantities evolve with time. Here, we are interested in a time function that describes the amount of alcohol in the body as time passes since consumption. In our scenario, we set time, \( t \), which starts counting from when alcohol enters the system.
In differential equations, the function of time usually shows how a variable such as alcohol content changes. For alcohol metabolism, time translates into constant change, which can be succinctly captured in our differential equation:

• \( A(t) \) represents the amount of alcohol at any given time \( t \).
• The equation \( \frac{dA}{dt} = -1 \) captures a simple relationship of how alcohol decreases over time, which is constant per unit time.

Think of time functioning as the foundation of the whole equation. It provides the framework over which changes occur and helps us analyze and predict how much alcohol remains in the body at any given hour based on initial consumption. This teaches students to appreciate the critical role time functions have when setting up and solving differential equations, reinforcing the importance of initial conditions in predicting future behavior.