Question

An environmental study indicates that the average daily level \(P\) of a certain pollutant in the air, in parts per million, can be modeled by the equation \(P=0.25 \sqrt{0.5 n^{2}+5 n+25}\) where \(n\) is the number of residents of the community, in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is 12,000 .

Short answer

To find the rate at which the level of pollutant is increasing when the population is 12,000, differentiate the given function with respect to population and evaluate at the given point. Calculate the derivative and substitute \(n=12\) to find the rate of change of pollutant level at this point.

Step-by-step solution

Differentiate the Function

First, differentiate \(P\) with respect to \(n\). The derivative will allow to find the rate of change of the pollutant level. Due to the structure of the equation, the Chain Rule will need to be applied. Given \(P=0.25 \sqrt{0.5 n^{2}+5 n+25}\), rewrite this as \(P=0.25 (0.5 n^{2}+5 n+25)^{0.5}\). Then take the derivative, \(P'\), of \(P\) with respect to \(n\).

Apply Chain Rule

Applying the Chain Rule, the derivative \(P'\) is \(0.25 * 0.5 * (0.5 n^{2}+5 n+25)^{-0.5} * (n^{2}+10n)\), simplifying gives \(P' =0.125(1/( \sqrt{0.5 n^{2}+5 n+25}))*(n^{2}+10n)\).

Substitute the given population value

With \(n=12\), substitute into the derived equation to find the rate of change of the pollutant level. The resultant value of \(P'\) at \(n=12\) is the rate at which the level of pollutant is increasing when the community has 12,000 residents.

Key concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of rates at which quantities change. It's primarily concerned with the concept of the derivative, which measures how a function changes as its input changes. This is especially useful in real-world problems where we want to understand the behavior of physical quantities that vary continuously.

When dealing with functions like the pollutant level in an environment, differential calculus helps us understand how quickly this level changes in response to changes in another variable, such as the population of a community. The utility of calculus becomes evident when we can predict and analyze dynamic systems, enabling us to make informed decisions or put in place appropriate measures.

Chain Rule

The chain rule is a fundamental theorem in calculus that is used for finding the derivative of a composite function. It essentially says that if you have two functions which are composed together, the derivative of the combined function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

In practice, when you encounter a function within a function, as in our pollutant level equation, the chain rule becomes essential for differentiation. The outer function in our case is the square root, while the inner function is the quadratic polynomial related to the population size. Without the chain rule, finding the rate at which the level of pollutant changes with respect to the population would be much more complicated, if not impossible.

Modeling with Algebra

Modeling with algebra involves representing real-world phenomena with algebraic equations and expressions. By creating a model, you translate a situation into a mathematical framework, which allows for analysis and predictions using mathematical tools. In environmental studies, such modeling is pivotal for quantifying relationships among various factors, like the correlation between the population size and pollutant level as illustrated by our problem.

The equation we have, expressing pollutant level as a function of population, is a result of simplifying complex relationships into an algebraic form that can be easily manipulated and understood. With algebra, we're equipped to transition from abstract mathematical expressions to meaningful interpretations of real-world behavior.

Environmental Mathematics

Environmental mathematics is an interdisciplinary field that applies mathematical methods to environmental issues. It encompasses the use of statistics, calculus, and modeling to address problems found in ecology, conservation, and resource management. This discipline is crucial in understanding and predicting environmental phenomena, from air pollution levels to the spread of invasive species.

In our textbook exercise, environmental mathematics comes into play as we seek a quantitative measure of the impact of population increase on pollutant levels in the air. By integrating algebraic modeling with differential calculus, we can derive concrete insights into how the environment reacts to human factors, which, in turn, aids in creating guidelines for sustainable development and environmental protection.