Problem 86
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 "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 \(\mathrm{mi} / \mathrm{hr}\) ) and \(v \geq 1 .\) If \(T=3\) and \(v=5,\) find \(P.\)
Step-by-Step Solution
Verified Answer
The population \( P \) is approximately 517,771.
1Step 1: Understand the Formula
We are given the formula that describes the temperature difference: \( T = \frac{0.25 P^{1/4}}{\sqrt{v}} \). We need to find \( P \) given that \( T = 3 \), \( v = 5 \).
2Step 2: Substitute Known Values
Substitute \( T = 3 \) and \( v = 5 \) into the equation: \( 3 = \frac{0.25 P^{1/4}}{\sqrt{5}} \).
3Step 3: Simplify the Equation
Multiply both sides by \( \sqrt{5} \) to eliminate the denominator: \( 3\sqrt{5} = 0.25 P^{1/4} \).
4Step 4: Solve for \( P^{1/4} \)
Divide both sides by \( 0.25 \) to solve for \( P^{1/4} \): \( P^{1/4} = \frac{3\sqrt{5}}{0.25} \). Calculate \( \frac{3\sqrt{5}}{0.25} = 3\sqrt{5} \times 4 = 12\sqrt{5} \).
5Step 5: Calculate \( \sqrt{5} \)
Calculate \( \sqrt{5} \approx 2.236 \). So, \( 12\sqrt{5} \approx 12 \times 2.236 = 26.832 \).
6Step 6: Calculate \( P \)
To find \( P \), raise \( 26.832 \) to the power of 4: \( P = (26.832)^4 \). This results in \( P \approx 517,771 \).
Key Concepts
Urban heat islandTemperature difference formulaAlgebraic substitutionEquation simplification
Urban heat island
Urban heat islands occur when urban areas have significantly higher temperatures compared to their rural counterparts. This is primarily due to human activities and the materials used in construction. Buildings, asphalt, and concrete absorb and retain heat, leading to warming.
Cities tend to have less vegetation compared to rural areas. This lack of greenery contributes to increased temperatures, as plants have a cooling effect. Reduced vegetation means less shade and less evaporation of water, both of which help to keep areas cool.
Urban heat islands can affect local weather. Higher temperatures can increase discomfort for city residents, influence energy consumption for cooling, and even affect local weather patterns.
Cities tend to have less vegetation compared to rural areas. This lack of greenery contributes to increased temperatures, as plants have a cooling effect. Reduced vegetation means less shade and less evaporation of water, both of which help to keep areas cool.
Urban heat islands can affect local weather. Higher temperatures can increase discomfort for city residents, influence energy consumption for cooling, and even affect local weather patterns.
Temperature difference formula
The temperature difference formula calculates the temperature difference between urban and rural areas. In this context, the formula is given by:
\[ T = \frac{0.25 P^{1/4}}{\sqrt{v}} \]
Where:
\[ T = \frac{0.25 P^{1/4}}{\sqrt{v}} \]
Where:
- \(T\) is the temperature difference in degrees Celsius
- \(P\) is the population of the urban area
- \(v\) is the average wind speed in miles per hour, with the condition \(v \geq 1\)
Algebraic substitution
Algebraic substitution is a technique used to replace variables in an equation with known values to solve the equation for an unknown variable.
In our problem, we started with the formula \( T = \frac{0.25 P^{1/4}}{\sqrt{v}} \). Given that \( T = 3 \) and \( v = 5 \), substitutions are made directly into the formula. The equation becomes:
\[ 3 = \frac{0.25 P^{1/4}}{\sqrt{5}} \]
By substituting these values, you simplify your calculations and eliminate the need for extra steps. This technique is commonly used to isolate and solve for an unknown variable efficiently.
In our problem, we started with the formula \( T = \frac{0.25 P^{1/4}}{\sqrt{v}} \). Given that \( T = 3 \) and \( v = 5 \), substitutions are made directly into the formula. The equation becomes:
\[ 3 = \frac{0.25 P^{1/4}}{\sqrt{5}} \]
By substituting these values, you simplify your calculations and eliminate the need for extra steps. This technique is commonly used to isolate and solve for an unknown variable efficiently.
Equation simplification
Equation simplification involves reducing an equation to a simpler form while keeping the equality intact.
After substitution, the next step is to clear fractions by multiplying through by the denominator, leading to:
\[ 3\sqrt{5} = 0.25 P^{1/4} \]
This gets the variable expression on one side. By isolating \(P^{1/4}\), you'll use operations such as division. Dividing both sides by 0.25 simplifies the equation further:
\[ P^{1/4} = 12\sqrt{5} \]
Each calculation step keeps transforming the equation into simpler forms until you solve for the desired variable, \(P = (26.832)^4\), producing an approximate population. This systematic approach ensures accuracy and clarity in problem-solving.
After substitution, the next step is to clear fractions by multiplying through by the denominator, leading to:
\[ 3\sqrt{5} = 0.25 P^{1/4} \]
This gets the variable expression on one side. By isolating \(P^{1/4}\), you'll use operations such as division. Dividing both sides by 0.25 simplifies the equation further:
\[ P^{1/4} = 12\sqrt{5} \]
Each calculation step keeps transforming the equation into simpler forms until you solve for the desired variable, \(P = (26.832)^4\), producing an approximate population. This systematic approach ensures accuracy and clarity in problem-solving.
Other exercises in this chapter
Problem 85
Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your a
View solution Problem 85
Simplify the expression. $$\frac{\left(x^{2}+2\right)^{3}(2 x)-x^{2}(3)\left(x^{2}+2\right)^{2}(2 x)}{\left[\left(x^{2}+2\right)^{3}\right]^{2}}$$
View solution Problem 86
Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your a
View solution Problem 86
Simplify the expression. $$\frac{\left(x^{2}-5\right)^{4}\left(3 x^{2}\right)-x^{3}(4)\left(x^{2}-5\right)^{3}(2 x)}{\left[\left(x^{2}-5\right)^{4}\right]^{2}}$
View solution