Problem 58 - Brief Applied Calculus

Question

ENVIRONMENTAL SCIENCE: Water Quality Downstream from a waste treatment plant the amount of dissolved oxygen in the water usually decreases for some distance (due to bacteria consuming the oxygen) and then increases (due to natural purification). A graph of the dissolved oxygen at various distances downstream looks like the curve below (known as the "oxygen sag"). The amount of dissolved oxygen is usually taken as a measure of the health of the river. Suppose that the amount of dissolved oxygen \(x\) miles downstream is \(D(x)=0.2 x^{2}-2 x+10 \mathrm{mpl}\) (milligrams per liter) for \(0 \leq x \leq 20\). Use this formula to find the instantaneous rate of change of the dissolved oxygen: a. 1 mile downstream. b. 10 miles downstream. Interpret the signs of your answers.

Step-by-Step Solution

Answer

-1.6 mg/L at 1 mile, indicating decrease; 2 mg/L at 10 miles, indicating increase.

Step 1: Understand the Problem

We need to determine the instantaneous rate of change of dissolved oxygen at different distances downstream. This involves using the derivative of the function that represents dissolved oxygen levels, \( D(x) = 0.2x^2 - 2x + 10 \), where \( x \) is the distance downstream in miles.

Step 2: Find the Derivative of D(x)

To find the instantaneous rate of change, we differentiate the function \( D(x) = 0.2x^2 - 2x + 10 \) with respect to \( x \). The derivative is \( D'(x) = 0.4x - 2 \). This derivative will be used to find the rate of change at specific distances.

Step 3: Calculate the Derivative at 1 Mile Downstream

Substitute \( x = 1 \) into the derivative \( D'(x) = 0.4x - 2 \) to find the rate of change at 1 mile downstream.\[D'(1) = 0.4(1) - 2 = 0.4 - 2 = -1.6\]

Step 4: Calculate the Derivative at 10 Miles Downstream

Substitute \( x = 10 \) into the derivative \( D'(x) = 0.4x - 2 \) to find the rate of change at 10 miles downstream.\[D'(10) = 0.4(10) - 2 = 4 - 2 = 2\]

Step 5: Interpret the Results

At 1 mile downstream, the rate of change is -1.6 mg/L per mile, indicating the dissolved oxygen level decreases. At 10 miles downstream, the rate of change is 2 mg/L per mile, indicating the dissolved oxygen level increases.

Key Concepts

Dissolved Oxygen

Dissolved oxygen (DO) is a critical factor in evaluating the quality and health of water bodies such as rivers and lakes. It refers to the amount of oxygen that is present in the water, which is essential for the survival of aquatic life.
Water with high levels of dissolved oxygen is generally considered healthy, whereas low levels can be harmful to fish and other organisms. The amount of dissolved oxygen can fluctuate due to several factors:

• Water temperature: Colder water can hold more dissolved oxygen than warmer water.
• Presence of organic matter: Decomposing organic matter can reduce DO levels as bacteria consume oxygen in the process of breaking it down.
• Photosynthesis: Aquatic plants and algae produce oxygen during the day through photosynthesis, increasing DO levels.
• Flowing water: Moving water tends to contain more dissolved oxygen than stagnant water.

In the context of the original exercise, analyzing the change in dissolved oxygen downstream from a waste treatment plant can help us understand how pollution impacts water quality.
The mathematical model given in the exercise allows us to predict changes in DO levels over a certain distance to gauge the health of the river's ecosystem.

Environmental Science

Environmental science is an interdisciplinary field that studies the interactions between physical, chemical, and biological components of the environment. Its goal is to understand how natural and human-induced processes affect ecosystems.
This field employs tools and knowledge from various scientific disciplines such as biology, chemistry, physics, geology, and sociology to apply solutions to environmental challenges. Understanding dissolved oxygen levels falls under the realm of environmental science, as these levels are an integral measurement of water quality. In the exercise, the focus is on the effects of a waste treatment plant on the river ecosystem:

• Pollution sources: Waste treatment plants can introduce organic materials into the water that affects DO levels.
• Ecological impact: Changes in dissolved oxygen can significantly affect aquatic life, as many species are sensitive to DO levels.
• Monitoring and data collection: Analyzing DO levels helps environmental scientists assess the effectiveness of pollution control measures and plant operations.

By combining mathematical models with real-world data, environmental scientists can predict outcomes, inform policy decisions, and propose strategies to preserve aquatic ecosystems.

Derivative

In mathematics, a derivative represents the rate of change of a function concerning one of its variables. In simpler terms, it tells us how a function's output changes as its input changes.
The concept of the derivative is fundamental in calculus and is widely used in various fields, including physics, engineering, and environmental science.In the context of the exercise, the derivative helps to determine the instantaneous rate of change of dissolved oxygen levels at different points downstream:

• The given function was: \(D(x) = 0.2x^2 - 2x + 10\).
• By differentiating \(D(x)\), we find \(D'(x) = 0.4x - 2\).
• This new function, \(D'(x)\), gives us the rate of change of dissolved oxygen at any point \(x\) in the river.
• Substituting specific values into \(D'(x)\) provides insights into whether the dissolved oxygen is increasing or decreasing at that specific location.

The interpretation of these calculated values is crucial: A negative derivative at 1 mile downstream indicates a decrease in DO, while a positive derivative at 10 miles downstream signals an increase.
This mathematical approach allows scientists to identify problem areas in the river system and evaluate the environmental impacts of human activities.