Problem 7 - Applied Calculus

Question

A pollutant spilled on the ground decays at a rate of \(8 \%\) a day. In addition, clean-up crews remove the pollutant at a rate of 30 gallons a day. Write a differential equation for the amount of pollutant, \(P\), in gallons, left after \(t\) days.

Step-by-Step Solution

Answer

\( \frac{dP}{dt} = -0.08P - 30 \)

Step 1: Understand the problem

The problem describes a scenario where a pollutant is decreasing in two ways: it decays by 8% per day naturally, and a cleanup crew removes 30 gallons per day. We need to represent how the pollutant decreases with time mathematically using a differential equation.

Step 2: Express the decay rate as a differential expression

The decay of the pollutant is described as 8% per day. This means the rate of change of the pollutant due to decay is proportional to the current amount of the pollutant. Mathematically, this is expressed as \( -0.08P \), where \( P \) is the amount of pollutant.

Step 3: Account for the removal by the cleanup crew

In addition to decaying, the pollutant is being removed at a constant rate of 30 gallons per day. This is expressed as \( -30 \) in the differential equation, as this is a constant subtractive rate from the current amount.

Step 4: Combine the two rates into a differential equation

To form the differential equation, combine the natural decay rate and the cleanup rate into a single expression describing the rate of change of the pollutant over time, \( \frac{dP}{dt} \). This gives us \( \frac{dP}{dt} = -0.08P - 30 \).

Key Concepts

Pollution Decay

Pollution decay refers to the reduction of pollutant substances over time. In our scenario, this process occurs naturally as the pollutant breaks down or dissipates through chemical reactions and environmental factors. We are told that the spill decays at a rate of 8% per day. To mathematically describe this phenomenon, we use a concept called *exponential decay*. This means that the amount of pollutant decreases by a percentage of its current amount each day. The expression representing just this natural decay is given by \[-0.08P\]where \(P\) is the amount of pollutant at any time \(t\). Understanding pollution decay helps us to predict how quickly a contaminant will decrease over time and what interventions might be necessary to hasten this process.

Rate of Change

The rate of change is a fundamental concept in calculus that describes how a quantity changes over time. In this exercise, we have two rates that contribute to the overall rate of change in the amount of pollutant:

• The natural decay, which is an 8% reduction of the current amount per day.
• The removal by cleanup efforts, which subtracts 30 gallons each day.

These two components are combined to express the rate at which the pollutant decreases over time. The overall rate of change is given by the differential equation:\[\frac{dP}{dt} = -0.08P - 30\]Here, \( \frac{dP}{dt} \) represents the rate of change of the pollutant \( P \) over time \( t \). Negative signs indicate a decrease, aligning with both the natural decay and human-mediated removal processes.

Exponential Decay

Exponential decay is a type of decay where the rate is proportional to the current value. It is a key concept in modeling pollutants as it helps us understand how pollutants diminish over time under natural conditions.In the context of the problem, exponential decay is modeled by the term \(-0.08P\). The negative sign indicates a reduction, while the coefficient -0.08 reflects an 8% daily decrease in the pollutant's amount. This concept is particularly useful because it not only describes the natural decay process, but also applies to any substance or quantity that decreases at a rate proportional to its size:

• Radioactive decay
• Half-life of drugs in the body
• Depreciation of an asset

Exponential decay forms the backbone of many predictions and calculations in environmental science, finance, and medicine, making it a versatile and powerful tool in various fields.