Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 $$ A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \quad(0 \leq t \leq 11) $$ 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.

Step-by-Step Solution

Answer

The pollution is at its highest level at 11:30 a.m. with a pollutant standard index (PSI) of 164.

Step 1: Differentiate the function

To find the maximum level of pollution, we need to find the critical points of the function A(t). Critical points occur where the derivative equals zero. Therefore, we'll start by finding the derivative of A(t) with respect to t. Given function, \(A(t) = \frac{136}{1+0.25(t-4.5)^{2}} + 28 \) Differentiating \(A(t)\): \(\frac{dA(t)}{dt} = -\frac{136(0.25)(2)(t-4.5)}{(1+0.25(t-4.5)^{2})^2}\)

Step 2: Find the critical points

Now, let's find the critical points by setting the derivative equal to zero and solving for t: \(-\frac{136(0.25)(2)(t-4.5)}{(1+0.25(t-4.5)^2)^2} = 0 \) The fraction equals zero when its numerator equals zero. So, we can solve for t. \((0.25)(2)(t - 4.5) = 0\) Divide both sides by 0.5: \((t - 4.5) = 0\) Solve for t: \(t = 4.5\) The critical point is at t = 4.5 hours.

Step 3: Evaluate the function at the critical point and endpoints

To find the maximum value for A(t), we need to check the value of the function A(t) at the critical point (t = 4.5) as well as the endpoints (t = 0 and t = 11). \(A(0) = \frac{136}{1+0.25(0-4.5)^{2}} + 28\) \(A(4.5) = \frac{136}{1+0.25(4.5-4.5)^2} + 28\) \(A(11) = \frac{136}{1+0.25(11-4.5)^2} + 28\) Calculating the values: \(A(0) \approx 49.944\) \(A(4.5) = 164\) \(A(11) \approx 47.44\)

Step 4: Determine the maximum value

Comparing the function values at the critical point and endpoints, it is clear that the maximum value of the function A(t) happens at t = 4.5 with A(4.5) = 164.

Step 5: Determine the time of the day when the pollution is at its highest level

Since t is measured in hours and t = 0 corresponds to 7 a.m., we can determine the time of the day when the pollution is at its highest level: 7 a.m. + 4.5 hours = 11:30 a.m. Thus, the pollution is at its highest level at 11:30 a.m.

Key Concepts

Critical Points

Critical points in a mathematical function are where the derivative of the function equals zero or is undefined. They are important because these points can indicate potential maximum or minimum values, known as extrema, or points of inflection.

In the context of environmental mathematics, especially when dealing with pollution indices like nitrogen dioxide levels, finding the critical points helps us determine when the pollution level is highest. This is key for planning activities and implementing pollution control measures.

To find critical points in the function given, we set the derivative \( \frac{dA(t)}{dt} \) to zero. This simplifies to finding when the numerator \( (t - 4.5) = 0 \), which solves when \( t = 4.5 \).

After identifying this critical point, we compared it against endpoint values in the given interval (0 to 11 hours) to confirm if it truly represents the highest level of pollution.

Differentiation

Differentiation is a technique in calculus used to find the rate at which a quantity changes. It is fundamental in finding the critical points in functions. The derivative of a function essentially tells us the slope of the tangent line to the function's curve at any given point.

In the provided problem, the function \( A(t) = \frac{136}{1+0.25(t-4.5)^{2}} + 28 \) models nitrogen dioxide levels over time. We differentiated it to find when the pollutant levels change.

This process involves using rules of differentiation on the rational function, paying special attention to the chain and quotient rules. Calculating the derivative gives us useful information to locate when the speed of pollution increase is zero, indicating a possible peak or trough. Thus, calculus aids environmental scientists by providing tools to predict and analyze environmental data trends.

Pollutant Standard Index

The Pollutant Standard Index (PSI) is a numerical scale that measures air quality based on the concentration of pollutants. It is widely used to communicate the health effects of air pollution to the public.

In the problem, nitrogen dioxide levels are expressed in PSI, helping relate the data to a standardized health impact scale. Generally, PSI values:

• 0 - 50: Good
• 51 - 100: Moderate
• 101 - 150: Unhealthy for sensitive groups
• 151 - 200: Unhealthy
• 201 - 300: Very unhealthy
• Above 301: Hazardous

In Long Beach, by identifying when the maximum PSI occurs (164 at 4.5 hours), residents can be advised when it might be safest to minimize outdoor exposure. Awareness of PSI levels allows communities worldwide to make informed decisions about environmental health impacts.

Environmental Mathematics

Environmental Mathematics integrates mathematical principles and models to solve and understand environmental issues. By using mathematical formulas and methods, environmental scientists can measure and predict environmental conditions, such as air quality.

The problem demonstrates the application of a mathematical model to estimate nitrogen dioxide pollution levels over time in Long Beach. Applying differentiation and critical points highlights how mathematical techniques assist in identifying significant events or periods, like peak pollution times.

As environmental challenges like air pollution grow, these math models become even more vital. They enable policymakers to prepare and respond effectively to fluctuating environmental conditions. Additionally, understanding data trends through Environmental Mathematics promotes a better grasp of human impacts on ecosystems, guiding sustainable practices.