Air Pollution The amount of nitrogen dioxide, a brown gas...

Air Pollution The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by $$A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \qquad(0 \leq t \leq 11)$$ where $A(t)$ is measured in pollutant standard index (PSI) and $t$ is measured in hours, with $t=0$ corresponding to 7 A.M. Determine the time of day when the pollution is at its highest level.

Air Pollution The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by
$$A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \qquad(0 \leq t \leq 11)$$
where $A(t)$ is measured in pollutant standard index (PSI) and $t$ is measured in hours, with $t=0$ corresponding to 7 A.M. Determine the time of day when the pollution is at its highest level.

Key Concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. They frequently appear in real-life models where relationships between quantities are complex yet can be simplified by polynomial division. The structure of the numerator and denominator in rational functions can provide insights into the behavior of the function, such as asymptotic behavior or local extrema, helping to illuminate how certain variables influence the overall output.

Function Transformation

Function transformation involves shifting, stretching, or compressing the graph of a function to simplify its analysis and interpretation. Recognizing such transformations, such as horizontal shifts indicated by expressions like (t - constant), enables one to identify key characteristics of the function without having to perform complex calculations. This is particularly useful when analyzing the behavior of a function to determine its maximum or minimum values.

Optimization

Optimization is the mathematical process of finding the maximum or minimum value of a function under given constraints. In many applied problems, such as determining peak pollution levels, optimization techniques are used to identify the point at which a particular function attains its extreme value. This often involves analyzing the structure of the function or computing its derivative to locate critical points and evaluating these along with the endpoints of the domain.

Critical Points

Critical points are values of the variable where the derivative of a function either vanishes or is undefined, and they often correspond to local maxima or minima. In the context of optimization problems, identifying these points is crucial because they indicate where the function might reach extreme values. By analyzing these critical points within the domain of interest, one can determine where the phenomenon being modeled — for example, an increase or decrease in pollutant levels — is at its most pronounced.

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