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 need to write a differential equation that describes how the alcohol is metabolized and excreted. We are told alcohol leaves the body at a constant rate of one ounce per hour.

Step 2: Define the rate of change

Since alcohol is excreted at a constant rate of one ounce per hour, the rate of change of the alcohol content in the body can be expressed as a constant rate of decrease. Mathematically, this can be described by the differential equation \( \frac{dA}{dt} = -1 \).

Step 3: Interpret the differential equation

The differential equation \( \frac{dA}{dt} = -1 \) tells us that the amount of alcohol in the body decreases by one ounce per hour. This constant rate reflects the continuous metabolism and excretion of alcohol.

Key Concepts

Rate of Change

The concept of "rate of change" is central to understanding how quantities evolve over time. In the realm of differential equations, a rate of change is often represented by a derivative. In this problem, we are interested in the rate at which alcohol is metabolized and excreted from the body.
The rate of change can be thought of as the speed at which a certain quantity, like alcohol, is decreasing over time. Here, you might express this mathematically as \(\frac{dA}{dt}\), where \(A\) is the amount of alcohol remaining in the body, and \(t\) represents time in hours.

• The negative sign in the differential equation \(\frac{dA}{dt} = -1\) indicates a decrease in the alcohol content.
• The constant \(-1\) shows that the change is uniform, meaning one ounce of alcohol is metabolized and excreted from the body every hour.

Understanding this equation means understanding that this is a linear problem: the rate at which alcohol leaves the body is constant over time, and thus the relationship is predictably simple.

Metabolism

Metabolism refers to all the biochemical processes that happen within a living organism to maintain life. In this context, we are primarily concerned with how the body processes alcohol.
When alcohol is consumed, the body works to break it down and remove it. This involves a complex series of pathways primarily situated in the liver, where enzymes play a critical role.

• Liver enzymes, primarily alcohol dehydrogenase (ADH), convert alcohol into acetaldehyde.
• Acetaldehyde is further metabolized into acetate, which is eventually broken down into carbon dioxide and water.
• This metabolic process is relatively constant, leading to the constant rate mentioned in the problem.

The steady nature of this metabolic pathway is why the differential equation \(\frac{dA}{dt} = -1\) accurately captures the essence of how alcohol is processed by the body in one-hour increments.

Excretion

Excretion is the process of removing waste and excess substances from the body. When alcohol is consumed, it eventually needs to be excreted. This happens after it has been metabolized into simpler compounds that the body can either use or eliminate.
In the body, excretion of alcohol or its metabolites occurs through several pathways:

• Primarily through metabolism in the liver, as previously explained.
• A small percentage is excreted unmetabolized, through breath, sweat, and urine.
• Finally, the majority is ultimately processed by the kidneys and excreted via urine.

This process is vital because it prevents the accumulation of toxic levels of alcohol in the bloodstream. The differential equation \(\frac{dA}{dt} = -1\) captures this excretion process, reflecting the steady removal rate that prevents alcohol buildup and supports body homeostasis.