Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

Population density The population density \(D\) (in people/ \(\mathrm{mi}^{2}\) ) in a large city is related to the distance \(x\) (in miles) from the center of the city by $$ D=\frac{5000 x}{x^{2}+36} $$ (a) What happens to the density as the distance from the center of the city changes from 20 miles to 25 miles? (b) What eventually happens to the density? (c) In what areas of the city does the population density exceed 400 people/ \(\mathrm{mi}^{-2}\) ?

Step-by-Step Solution

Answer

(a) Density decreases from 229.36 to 189.86 as distance increases from 20 to 25 miles. (b) Density approaches 0 as distance goes to infinity. (c) Density exceeds 400 people/mi\(^2\) for 3.07 < x < 11.73.

Step 1: Understanding the Population Density Function

The given population density function is \( D = \frac{5000x}{x^2 + 36} \). This represents how population density varies with distance from the city's center.

Step 2: Calculate Density at 20 Miles

Substitute \( x = 20 \) into the equation: \[ D = \frac{5000 \times 20}{20^2 + 36} = \frac{100000}{400 + 36} = \frac{100000}{436} \approx 229.36 \] Thus, the population density at 20 miles is approximately 229.36 people/mi\(^2\).

Step 3: Calculate Density at 25 Miles

Substitute \( x = 25 \) into the equation: \[ D = \frac{5000 \times 25}{25^2 + 36} = \frac{125000}{625 + 36} = \frac{125000}{661} \approx 189.86 \] Thus, the population density at 25 miles is approximately 189.86 people/mi\(^2\).

Step 4: Determine Density as \( x \to \infty \)

Analyze the behavior of the density function as \( x \) approaches infinity. The function simplifies to: \[ D = \frac{5000x}{x^2 + 36} \approx \frac{5000x}{x^2} = \frac{5000}{x}\] As \( x \to \infty \), \( \frac{5000}{x} \to 0 \). Therefore, the density approaches zero as the distance increases indefinitely.

Step 5: Condition for Density Exceeding 400

Solve \( \frac{5000x}{x^2 + 36} > 400 \): Multiply both sides by \( x^2 + 36 \): \[ 5000x > 400(x^2 + 36) \] \[ 5000x > 400x^2 + 14400 \] Rearrange terms: \[ 400x^2 - 5000x + 14400 < 0 \] Solve this quadratic inequality for \( x \). The roots determine the range where the density exceeds 400. Using the quadratic formula: Roots are approximately \( x = 3.07 \) and \( x = 11.73 \). Thus, the density exceeds 400 people/mi\(^2\) for \( 3.07 < x < 11.73 \).

Key Concepts

Understanding Quadratic Inequality

Quadratic inequalities might feel intimidating at first, but they're quite manageable with practice. In mathematics, a quadratic inequality involves an inequality with a quadratic expression. For example, in the problem given, the inequality is:\[ 400x^2 - 5000x + 14400 < 0 \]This inequality is derived from our aim to find when the population density exceeds 400 people/miep2 by rearranging the given condition. To solve this, we first need to determine the roots of the associated quadratic equation:\[ 400x^2 - 5000x + 14400 = 0 \]We use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 400\), \(b = -5000\), and \(c = 14400\). Substituting these values gives us the roots, which then help locate the intervals where the inequality holds true. These roots determine the critical points, and by testing intervals between and beyond these roots, we find that the inequality is valid for the range \(3.07 < x < 11.73\). This means that in this interval, the population density surpasses 400 people/miep2.

Behavior at Infinity

Understanding how a population density function behaves as the distance from the center goes to infinity gives a sense of its long-term trend. When tackling this type of problem, we analyze the term \(D = \frac{5000x}{x^2 + 36}\). As \(x\) becomes very large, the \(x^2\) term in the denominator dominates the constant value.To evaluate \(\lim_{{x \to \infty}} \frac{5000x}{x^2 + 36}\), we can simplify the expression to:\[ D = \frac{5000x}{x^2} = \frac{5000}{x} \]This simplification occurs because the \(36\) becomes insignificant compared to \(x^2\). As \(x\) approaches infinity, \(\frac{5000}{x}\) approaches zero, indicating that population density decreases and eventually nears zero as distance from the city center increases indefinitely. This insight is crucial for understanding distribution patterns in expansive urban environments.

City Center Distance and Population Density

The concept of city center distance refers to the measurement of how far a point is from the center of the city. In city planning and population studies, it is crucial to understand that as you move away from the city center, several factors like infrastructure availability and accessibility could affect population density.In the problem, population density is expressed as a function of distance \(x\) from the center. Initially, as we move from the where distance is 0 to somewhere significantly farther like 20 miles, we observe the population density going through notable changes. For example, at 20 miles, density is calculated as 229.36 people/miep2, whereas at 25 miles it drops to 189.86 people/miep2. The decreasing trend likely reflects suburban areas where less clustering of people occurs compared to the city core.Therefore, understanding how population density changes with distance can help in city planning and efficient resource allocation.

Solving Equations in Context

Solving equations in mathematics, especially within real-world applications like population density, involves more than just manipulating numbers. It requires translating real-world problems into mathematical formats that can be analyzed. For instance, in determining areas where population density exceeds a certain threshold, we set up an inequality with the population density function. Then we derived the quadratic equation:\[ 400x^2 - 5000x + 14400 = 0 \]By using this quadratic equation, we find possible solutions that guide us in understanding the model's behavior. We substituted into the quadratic formula and solved, yielding roots that allow us to define intervals where our density condition is met.Equation solving often repeats these steps: formulating the problem, setting up equations, applying solution methods like the quadratic formula, and finally interpreting results back into the real-world scenario. This sequence helps uncover valuable insights into spatial population patterns.