Problem 10 - Applied Calculus

Question

Toxins in pesticides can get into the food chain and accumulate in the body. A person consumes 10 micrograms a day of a toxin, ingested throughout the day. The toxin leaves the body at a continuous rate of \(3 \%\) every day. Write a differential equation for the amount of toxin, \(A\), in micrograms, in the person's body as a function of the number of days, \(t\).

Step-by-Step Solution

Answer

\( \frac{dA}{dt} = 10 - 0.03A(t) \) is the differential equation.

Step 1: Define Variables

Let \( A(t) \) represent the amount of toxin in the body in micrograms as a function of time \( t \) in days. The constant ingestion rate of the toxin is 10 micrograms per day.

Step 2: Analyze Toxin Rate of Change

There are two components affecting the rate of change of toxin: the intake rate, which adds 10 micrograms per day, and the elimination rate, which removes \(3\%\) of the toxin present in the body per day.

Step 3: Formulate the Differential Equation

The rate of change of the toxin is given by the sum of the intake rate and the elimination rate. The elimination rate is \(-0.03A(t)\) since \(3\%\) of the current amount \(A(t)\) is being lost each day. Therefore, the differential equation is: \[ \frac{dA}{dt} = 10 - 0.03A(t) \]

Step 4: Write Final Equation

Combine both effects into a single differential equation to represent the change in toxin amount over time. The formulated equation is as follows: \[ \frac{dA}{dt} = 10 - 0.03A(t) \]

Key Concepts

Toxin Accumulation

When we talk about toxin accumulation, we refer to the buildup of harmful substances in the body over time. In our exercise, the accumulation of a toxin from pesticide exposure occurs because of daily consumption. The person ingests 10 micrograms of this toxin every day. That might seem small, but over days and weeks, it is a continuous stream into the body, creating a cumulative effect.
This buildup happens because the rate at which the toxin is being introduced into the body is constant, while the body's elimination process may not be as efficient to remove it entirely, leading to accumulation. If the elimination isn't fast enough to counteract intake, toxins can build up to dangerous levels over time.

• The daily intake of toxin is consistent, at 10 micrograms per day.
• Accumulation occurs because more toxin enters than leaves the body daily.

Rate of Change

The rate of change in the context of differential equations is all about how a quantity evolves over time. For toxins in the body, this means examining how the amount of toxin increases or decreases each day. We look at two main factors affecting this rate: the daily intake of toxins and the body's ability to eliminate them.
A differential equation helps express this relationship mathematically by showing how the amount of toxin in the body changes with time. This is achieved by combining both the intake of new toxins and the elimination of existing ones in the body.

• The rate of toxin entering the body is +10 micrograms per day.
• The rate of toxin being eliminated from the body is counted negatively.
• This relationship is captured in the equation: \( \frac{dA}{dt} = 10 - 0.03A(t) \).

Body Elimination Rate

The body elimination rate describes how quickly the body can remove toxins to prevent them from accumulating to harmful levels. In our exercise, the elimination rate is given as a continuous 3% per day. This means that 3% of the toxin present in the body is naturally eliminated each day without additional action from the person.
This rate can significantly impact how quickly or slowly toxins accumulate. If the consumption rate is higher than the elimination rate, the toxin can accumulate. On the contrary, if the body eliminates toxins faster than they're consumed, the accumulation will decrease over time.

• The elimination rate is 3% of whatever is currently in the body.
• Mathematically, this is represented as a loss of 0.03 times the current amount \(A(t)\) each day.
• This is a crucial aspect of keeping toxin accumulation in check.