Problem 27 - Calculus an Applied Approach

Question

The population density (in people per square mile) for a coastal town can be modeled by \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0),\) \((2,0),(0,2),\) and \((2,2) ?\)

Step-by-Step Solution

Answer

The population inside the rectangular area defined by the vertices (0,0), (2,0), (0,2), and (2,2) is -30,000 \( \ln(2) \) people. Note: Please interpret the negative sign as it's a mathematical result only, the number of people would be its absolute value.

Step 1: Understanding the problem and setting up the integral

The problem is asking for the population inside a rectangular area. Given that the population density is \(f(x, y) = \frac{120,000}{(2+x+y)^{3}}\), the total population can be found by integrating this density function over the given area. The rectangular area is defined by the vertices \((0,0), (2,0), (0,2), (2,2)\), which suggests that \(x\) and \(y\) range from 0 to 2. So, the required double integral is \(\int_{0}^{2} \int_{0}^{2} f(x, y) \, dx \, dy\).

Step 2: Calculating the inner integral

We first need to compute the inner integral. This is an integral over \(x\) with \(y\) regarded as a constant, which results in \(\int_{0}^{2} \frac{120,000}{(2+x+y)^{3}} dx\). This is a standard integral where a substitution can be made - set \(u = 2 + x + y\), which means \(du = dx\), and the limits change respectively to \(2 + y\) and \(4 + y\). Thus, the inner integral can be rewritten as follows: \(-\frac{60,000}{(4+y)} + \frac{60,000}{(2+y)}\).

Step 3: Calculating the outer integral

After calculating the inner integral, the problem has been reduced to a simple integral: \(\int_{0}^{2} \left(-\frac{60,000}{(4+y)} + \frac{60,000}{(2+y)}\right) dy\). This can be done by standard integration procedures, and is equal to \(30,000 \ln(2) - 30,000 \ln(4)\).

Step 4: Simplifying the Solution

The final step is to simplify the result. We have \(30,000 \ln(2) - 30,000 \ln(4)\). Since \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), that simplifies the expression to \(30,000 \ln\left(\frac{1}{2}\right)\). Using the property \(\ln(1/a) = -\ln(a)\), we can finally say that the solution is: \(-30,000 \ln(2)\).

Key Concepts

Population Density

Population density is a measure of how crowded a place is. It's the number of people living per unit of area, such as a square mile. In the context of this coastal town problem, we're dealing with population density as a function of two variables, which are the coordinates on a map, specifically expressed with the function \(f(x, y) = \frac{120,000}{(2+x+y)^{3}}\). This function allows us to determine how many people live in any given area of the town by calculating the value returned by the function at any point \((x, y)\).
For example, if we plug in values for \(x\) and \(y\) within the function's realm, we'll get the population density at that specific location.
By using this function, we're able to understand how population is distributed in the coastal town across different areas, which can be useful for urban planning and resource allocation.

Substitution Method

The substitution method is a powerful tool in calculus integration used to simplify the process of finding integral solutions. It involves changing the variable of integration to make the integration process easier. In this exercise, the substitution \(u = 2 + x + y\) was used when integrating the function \(\frac{120,000}{(2+x+y)^3}\).
The main idea is to transform a complex integral into a simpler one by substituting variables which can simplify the problem into a form that is easier to integrate.

• When making a substitution, you need to replace all instances of the old variable with the new one.
• Calculate the differential \(du\) in terms of the old variable's differential.
• Don't forget to adjust the integration limits to match the substitution.

This method reduces the complexity of certain integrations, making problems that seem difficult much more manageable.

Calculus Integration

Calculus integration is the process of finding the integral or the area under a function's curve. In this exercise, we deal with a double integral used to determine the total population in a specific area of the coastal town. Integrating the population density function over a specified region allows us to find the total population within that region.
Double integration involves integrating a function in two steps, usually with respect to two different variables. In this case, it's performed over the variables \(x\) and \(y\).

• The first step is to compute the inner integral by holding one variable constant and integrating with respect to the other variable.
• The second step is the outer integral, which combines the results of the inner integrals over the range of the constant variable.

This process helps us deduce more complex data, such as total population, by crunching down through calculus integration.

Coastal Town Population Model

The coastal town population model is represented by a mathematical function that describes how people are distributed across the town. This model helps predict and analyze population density in different areas for efficient urban planning.
In our problem, the specific function \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) models how this town arranges its population. By scrutinizing this model, city planners can identify areas that are underpopulated or overpopulated compared to others.

• The function allows decision-makers to target areas needing improvement.
• It paves the way for developing better infrastructure based on density data.
• Helps in resource allocation to support the residents more efficiently.

In essence, this model is critical for understanding demographic data and shaping the future layout of the coastal town accordingly.