Problem 39 - Precalculus: Functions and Graphs

Question

As a result of urbanization, the temperatures in Paris have increased. In 1891 the average daily minimum and maximum temperatures were \(5.8^{\circ} \mathrm{C}\) and \(15.1^{\circ} \mathrm{C},\) respectively. Between 1891 and \(1968,\) these average temperatures rose \(0.019^{\circ} \mathrm{C} / \mathrm{yr}\) and \(0.011^{\circ} \mathrm{C} / \mathrm{yr},\) respectively. Assuming the increases were linear, find the year when the difference between the minimum and maximum temperatures was \(9^{\circ} \mathrm{C},\) and determine the corresponding average maximum temperature.

Step-by-Step Solution

Answer

The year is 1928.5, with an average maximum temperature of 15.51°C.

Step 1: Understand the problem

We need to find the year when the difference between the minimum and maximum temperatures in Paris is exactly 9°C and determine the average maximum temperature for that year. Given the rates of temperature increase, we can use a linear model to solve this problem.

Step 2: Write linear equations for temperatures

The linear equations for the minimum and maximum temperatures are based on the starting values from 1891 and their respective increases per year.Let's set "x" as the number of years after 1891.- Minimum temperature equation: \( T_{\text{min}} = 5.8 + 0.019x \)- Maximum temperature equation: \( T_{\text{max}} = 15.1 + 0.011x \)These equations describe the linear increase in minimum and maximum temperatures over time.

Step 3: Set the equation for temperature difference

The difference between the maximum and minimum temperature needs to be 9°C. Set up the equation for the difference as follows:\[ (15.1 + 0.011x) - (5.8 + 0.019x) = 9 \]

Step 4: Solve the equation for x

Simplify and solve the equation from Step 3:\[ 15.1 - 5.8 + 0.011x - 0.019x = 9 \] \[ 9.3 - 0.008x = 9 \] \[ 0.3 = 0.008x \] \[ x = \frac{0.3}{0.008} \] \[ x = 37.5 \]This means 37.5 years after 1891.

Step 5: Determine the year and check for accuracy

Add the calculated \(x\) to the base year 1891 to find the specific year:\[ \ \] \[ \text{Year} = 1891 + 37.5 \] \[ \text{Year} = 1928.5 \] This indicates the middle of 1928.

Step 6: Calculate the maximum temperature for 1928.5

Using the value of \(x = 37.5\), substitute it back into the equation for maximum temperature:\[ T_{\text{max}} = 15.1 + 0.011 \times 37.5 \] \[ T_{\text{max}} = 15.1 + 0.4125 \] \[ T_{\text{max}} = 15.5125 \] The average maximum temperature for 1928.5 would be approximately \(15.51^{\circ} \mathrm{C}.\)

Key Concepts

Urbanization and Temperature Change

Urbanization often leads to a rise in urban temperatures, a phenomenon known as the urban heat island effect. This occurs because human activities, buildings, and man-made surfaces absorb and retain heat. Urban areas tend to have higher temperatures than their surrounding rural areas. In the case of Paris, between 1891 and 1968, urbanization coincided with a measurable increase in both minimum and maximum temperatures.

• Minimum temperatures rose by approximately 0.019°C per year.
• Maximum temperatures rose by approximately 0.011°C per year.

This trend highlights the importance of understanding how urban growth affects local and global climates. When planning cities, architects and city planners must consider these effects to mitigate temperature increases.

Temperature Difference Calculation

Calculating temperature differences is crucial for understanding various climate phenomena. In our problem, we were interested in finding out the year when the difference between minimum and maximum temperatures reached exactly 9°C in Paris.
Initially, we used linear equations based on historical temperature data from 1891:

• For minimum temperature: \( T_{\text{min}} = 5.8 + 0.019x \)
• For maximum temperature: \( T_{\text{max}} = 15.1 + 0.011x \)

The difference between these two equations was set to 9°C, resulting in the equation: \[ (15.1 + 0.011x) - (5.8 + 0.019x) = 9 \] By solving this equation, we determined the specific number of years after 1891 when this temperature difference occurred.

Solving Linear Systems

To find the year in which the temperature difference equaled 9°C, we solved a simple linear system. Linear systems involve equations that aim to find unknown variables through elimination or substitution.
In the exercise, the unknown variable was \( x \), representing the years after 1891. We reworked the equation: \[ 9.3 - 0.008x = 9 \] and simplified it to find \( x \):\[ 0.3 = 0.008x \]\[ x = \frac{0.3}{0.008} \]\[ x = 37.5 \]This calculation allowed us to determine that 37.5 years after 1891 marked the solution. Linear systems like this help us model and solve real-world problems by using a straightforward approach.

Historical Climate Data Analysis

Analyzing historical climate data helps experts understand long-term climate trends and changes. This analysis often uses temperature records over several decades to establish patterns and predict future changes. In our example, the data from 1891 onwards enabled us to track how temperatures in Paris evolved over time due to urbanization.
To calculate trends, we rely on:

• Collecting consistent and accurate historical temperature records.
• Applying mathematical models like linear equations to interpret this data.

By analyzing such data, scientists can improve climate models, inform policymaking, and guide urban planning strategies. This ensures better preparedness for future climate changes.