Problem 62 - Algebra and Trigonometry with Analytic Geometry

Question

Urban areas have higher average air temperatures than rural areas, as a result of the presence of buildings, asphalt, and concrete. This phenomenon has become known as the urban heat island. The temperature difference \(T\) (in \({ }^{\circ} \mathrm{C}\) ) between urban and rural areas near Montreal, with a population \(P\) between 1000 and \(1,000,000\), can be described by the formula \(T=0.25 P^{1 / 4} / \sqrt{v}\), where \(v\) is the average wind speed (in mi/hr) and \(v \geq 1\). If \(T=3\) and \(v=5\), find \(P\).

Step-by-Step Solution

Answer

The population \(P\) is approximately 517,850.

Step 1: Understand given values and equation

We are given the formula for temperature difference as \(T = \frac{0.25 P^{1/4}}{\sqrt{v}}\). We know that the temperature difference \(T\) is 3 and the average wind speed \(v\) is 5. These values need to be substituted into the formula.

Step 2: Substitute values into the equation

Substitute \(T = 3\) and \(v = 5\) into the equation: \(3 = \frac{0.25 P^{1/4}}{\sqrt{5}}\). This will isolate the variable \(P\) to be solved.

Step 3: Simplify the equation

Since \( \sqrt{5} \) is a constant, calculate it to simplify: \(3 = \frac{0.25 P^{1/4}}{2.236}\). This equation needs to be simplified by multiplying both sides by 2.236.

Step 4: Solve for \(P^{1/4}\)

Multiply both sides by 2.236 to isolate \(P^{1/4}\) : \[3 \times 2.236 = 0.25 P^{1/4}\]. Simplifying gives \[6.708 = 0.25 P^{1/4}\]. Now, divide both sides by 0.25.

Step 5: Solve \(P^{1/4}\)

Divide both sides by 0.25: \[P^{1/4} = \frac{6.708}{0.25}\], which simplifies to \[P^{1/4} = 26.832\].

Step 6: Solve for \(P\)

Raise both sides of the equation to the power of 4 to solve for \(P\): \[P = (26.832)^4\]. This results in \[P \approx 517850.44\].

Key Concepts

Temperature Difference Formula

The urban heat island effect is a phenomenon where urban regions exhibit higher temperatures than surrounding rural areas. This can largely be boiled down to the abundance of heat-retaining surfaces like concrete and asphalt. A mathematical model representing this temperature difference is crucial for understanding and predicting the urban heat island effect.In our exercise, the formula given is: \[ T = \frac{0.25 P^{1/4}}{\sqrt{v}} \] This formula considers three variables:

• T: Temperature difference in degrees Celsius
• P: Population of the urban area, varying between 1000 and 1,000,000
• v: Average wind speed in miles per hour, where \( v \geq 1 \)

The task involves using known values of \( T \) and \( v \) to solve for \( P \). The key here is isolating \( P^{1/4} \) and solving it step-by-step to ultimately find \( P \). This approach helps us understand how different factors such as population and wind speed influence the temperature difference.

Population and Temperature Relation

Population density plays a significant role in the urban heat island effect. With more people in an area, there's often more energy usage and heat generation from buildings and vehicles. This factor is represented in the equation by the term \( P^{1/4} \).The relationship between population and temperature difference can be understood as:

• As population \( P \) increases, the temperature difference \( T \) also increases, assuming wind speed remains constant.
• The population is taken to the power of \( 1/4 \), suggesting that the increase in temperature is not linear. Smaller populations will result in a lesser increment compared to larger populations.

This formula effectively models the impact of increasing urbanization on temperature anomalies between urban and rural areas. Understanding this relationship can help in planning urban developments and mitigating excessive heat accumulation.

Wind Speed Effect on Temperature

Wind speed is a crucial factor in determining how heat builds up in urban areas. In the given formula \( T = \frac{0.25 P^{1/4}}{\sqrt{v}} \), wind speed \( v \) appears in the denominator, showing its inverse relationship with temperature difference.Understanding this relationship involves:

• A higher wind speed \( v \) results in a lower temperature difference \( T \), as the wind disperses the accumulated heat more effectively.
• The presence of \( \sqrt{v} \) indicates that changes in wind speed have a diminishing effect. A large increase in wind speed results in a smaller change in \( T \).

By incorporating wind speed into our understanding of urban heat islands, we can better predict and manage the temperature variations in urban environments, leveraging natural elements to counteract heat retention.