Problem 67 - Brief Applied Calculus

Question

Pollution The carbon monoxide level in a city is predicted to be \(0.02 x^{3 / 2}+1\) ppm (parts per million), where \(x\) is the population in thousands. In \(t\) years the population of the city is predicted to be \(x(t)=12+2 t\) thousand people. Therefore, in \(t\) years the carbon monoxide level will be $$P(t)=0.02(12+2 t)^{3 / 2}+1 \quad \text { ppm }$$ Find \(P^{\prime}(2)\), the rate at which carbon monoxide pollution will be increasing in 2 years.

Step-by-Step Solution

Answer

The rate of increase is 0.24 ppm per year at \(t = 2\).

Step 1: Understand the problem

We are given a formula for carbon monoxide level in terms of population \((x)\) and a function \(x(t)\) that represents population in the city over time \((t)\). The task is to determine how fast the pollution level \(P(t)\) is increasing when \(t = 2\). This requires calculating the derivative \(P'(t)\) and then evaluating it at \(t = 2\).

Step 2: Express pollution level as a function of time

Use the provided population function \(x(t) = 12 + 2t\) to substitute into the expression for pollution level: \[ P(t) = 0.02(12 + 2t)^{3/2} + 1 \] Now, \(P\) is expressed as a function of time \(t\).

Step 3: Differentiate pollution level with respect to time

We need \(P'(t)\), the derivative of \(P(t)\). Use the chain rule to differentiate: First, let \(u = 12 + 2t\), so \(P(t) = 0.02u^{3/2} + 1\). Then the derivative \(\frac{du}{dt} = 2\). Now, apply the chain rule: \(\frac{dP}{dt} = 0.02 \cdot \frac{d}{dt}(u^{3/2}) = 0.02 \cdot \frac{3}{2} \cdot u^{1/2} \cdot \frac{du}{dt} = 0.02 \cdot \frac{3}{2} \cdot u^{1/2} \cdot 2\). Simplifying gives: \(P'(t) = 0.06u^{1/2}\).

Step 4: Evaluate the derivative at \(t = 2\)

Substitute \(t = 2\) to find the value of \(P'(2)\): First, evaluate \(u = 12 + 2 \times 2 = 16\). Then, plug \(u = 16\) into the derivative: \(P'(2) = 0.06(16)^{1/2}\). Calculate \(\sqrt{16} = 4\), then: \(P'(2) = 0.06 \cdot 4 = 0.24\).

Step 5: Interpret the result

The derivative \(P'(2) = 0.24\) indicates that, in 2 years, the carbon monoxide level in the city is increasing at a rate of 0.24 ppm per year.

Key Concepts

Chain Rule

The chain rule is a powerful differentiation tool used in calculus to find the derivative of a composite function. Consider a composite function as a function defined by another function, like how the pollution level, \(P(t)\), is determined by the population over time, \(x(t)\). The chain rule helps determine how quickly one function is changing in relation to another.

When you apply the chain rule, you essentially break down the problem into more manageable parts. Here’s a simple analogy: imagine a set of nested boxes where you want to move from the outermost one to the innermost one. With the chain rule, you take things step by step.

• Start with the outer function and find its derivative with respect to the inner function.
• Then, find the derivative of the inner function with respect to the variable of interest (in this case, time \(t\)).
• Multiply these derivatives together to find the overall rate of change.

In our exercise, we have the carbon monoxide level as a function of both population \(x\) and time \(t\). By using the chain rule, we can connect these complexities and find how fast the pollution levels are changing over time, which we did by finding \(P'(t)\).

Population Models

Population models are mathematical equations or functions that predict the future population given certain conditions. They are vital for understanding how populations grow and how factors like pollution might change as a result.

One simple form of a population model is linear growth, like in our exercise where the population grows linearly with time \(x(t) = 12 + 2t\). Here, the initial population at time \(t = 0\) is 12 thousand, and it increases by 2 thousand each year. Linear growth means the same amount of increase occurs over each equal time period.

• Linear models are easy to work with and provide simple predictions.
• They are best suited for short-term predictions where other complex factors (like migration, death rates) are not dominant.
• In real-world scenarios, more complex models like exponential growth might be more accurate for long-term predictions.

Understanding population models helps us see how changes in population numbers can affect other variables, such as pollution levels in our exercise.

Carbon Monoxide Levels

Carbon monoxide (CO) is a pollutant that affects air quality and can have serious health impacts. Monitoring and predicting its levels are crucial for city planning and public health.

In the exercise in question, we use a mathematical prediction model for carbon monoxide levels that depend on the population: \(P(t) = 0.02(12 + 2t)^{3/2} + 1\). This model provides insights into how pollution might change as more people reside in the city.

• The pollutant levels can increase significantly with population due to more vehicles, industry emissions, and other sources.
• Our derivative, \(P'(t)\), shows the rate of change of pollution levels, critical for noting how rapidly the environment might be affected.
• By calculating this rate at a specific time (e.g., \(t = 2\)), city planners can make informed decisions on environmental policies.

Having accurate models and being able to compute rates of change helps in managing pollution and ensuring sustainable urban growth.