Problem 7 - Applied Calculus

Question

A pollutant spilled on the ground decays at a rate of \(8 \%\) a day. In addition, cleanup 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

The differential equation is \( \frac{dP}{dt} = -0.08P - 30 \).

Step 1: Understand the Decay Process

The pollutant decreases at a rate of \(8\%\) per day. This means that each day, \(P\) gallons will reduce by \(0.08P\) gallons due to decay.

Step 2: Consider Removal by Cleanup Crews

Each day, cleanup crews remove a constant amount of \(30\) gallons of the pollutant.

Step 3: Express the Total Change in the Pollutant

Combine the decay and removal rates to express the total daily change. The decay contributes a change of \(-0.08P\) gallons per day, and the removal contributes \(-30\) gallons per day. Therefore, the rate of change of \(P\) per day is the sum of these two changes.

Step 4: Write the Differential Equation

The differential equation representing the total rate of change of the pollutant is given by: \[ \frac{dP}{dt} = -0.08P - 30 \] This equation models the combined effect of both natural decay and manual removal on the pollutant over time.

Key Concepts

Rate of Decay

In the context of pollutants, the term 'rate of decay' refers to the natural decrease of a pollutant's quantity over time due to chemical or biological processes. This is typically expressed as a percentage. In our scenario, the rate of decay is 8% per day. This means each day, the amount of the pollutant reduces by 8% of its current volume.

To find out how much the pollutant decreases per day, we calculate 8% of the existing amount. Suppose the current amount is represented by the variable \(P\). The daily reduction due to natural decay is calculated as \(0.08P\), where 0.08 is the decimal representation of 8%.

The concept is closely related to exponential decay often used in physics and chemistry, where a quantity decreases at a rate proportional to its current value. In differential equations, this idea is useful because it helps to model how pollutants, radioactivity, and other substances decrease over time.

Cleanup Process

The cleanup process is an additional method of reducing the amount of pollutant present in our environment. This process involves human efforts to physically or chemically remove pollutants. For this particular problem, cleanup crews actively remove 30 gallons of pollutant from the site every day.

This removal rate is constant and unlike the natural decay rate, it does not depend on the current size of the pollutant spill. Every day, regardless of how much pollutant remains, 30 gallons are removed.

In mathematical terms, the rate of removal is represented as a constant value that does not change with the pollutant quantity. In our differential equation, this would be denoted by simply subtracting 30 from the total amount of pollutant daily. This straightforward approach highlights the role of human intervention in managing pollution levels and complements the natural decay process.

Pollutant Reduction

Pollutant reduction is the combined result of different processes that work together to bring down the level of pollution. In this scenario, we observe two key factors leading to pollutant reduction: the natural decay process and cleanup efforts performed by humans.

Together, these factors are expressed mathematically through a differential equation. The equation \(\frac{dP}{dt} = -0.08P - 30\) illustrates how, every day, the total change in the pollutant amount is calculated by adding the effects of both processes - the 8% decay and the constant 30 gallons removal.

Differential equations are a powerful tool for modeling such dynamic systems where change over time is influenced by multiple factors. They allow us to predict how quickly the pollution will reduce and potentially reach safe levels. Understanding the roles and impacts of each component in the equation helps decision-makers strengthen pollutant reduction strategies and enhance their effectiveness.