Problem 41 - Algebra and Trigonometry with Analytic Geometry

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, and the average maximum temperature is approximately 15.51°C.

Step 1: Define Initial Conditions

The average minimum temperature in 1891 was \(5.8^{\circ} \mathrm{C}\) and the average maximum temperature was \(15.1^{\circ} \mathrm{C}\).

Step 2: Establish Rates of Change

Between 1891 and 1968, the average minimum temperature increased by \(0.019^{\circ} \mathrm{C} / \mathrm{yr}\), and the maximum temperature increased by \(0.011^{\circ} \mathrm{C} / \mathrm{yr}\).

Step 3: Express Future Temperatures in Terms of Time

Let \(x\) represent the number of years after 1891. The future minimum temperature \(T_{min}(x)\) is given by \(T_{min}(x) = 5.8 + 0.019x\). The future maximum temperature \(T_{max}(x)\) is given by \(T_{max}(x) = 15.1 + 0.011x\).

Step 4: Setup the Condition for Temperature Difference

We need to find the year \(x\) when the difference between the maximum and minimum temperatures is \(9^{\circ} \mathrm{C}\). Set up the equation: \(T_{max}(x) - T_{min}(x) = 9\).

Step 5: Solve the Equation for x

Substitute the expressions for \(T_{min}(x)\) and \(T_{max}(x)\) to get \((15.1 + 0.011x) - (5.8 + 0.019x) = 9\). Simplify to \(9.3 - 0.008x = 9\). Solve for \(x\) to find: \(0.008x = 0.3\), giving \(x = 37.5\).

Step 6: Determine the Year and Corresponding Maximum Temperature

Since \(x = 37.5\), this corresponds to the year 1891 + 37.5 = 1928.5. The corresponding average maximum temperature is \(T_{max}(37.5) = 15.1 + 0.011 \times 37.5 = 15.5115^{\circ} \mathrm{C}\).

Key Concepts

Temperature Change

Understanding temperature change is crucial when studying climate and environmental sciences. In Paris, over a period of time due to urbanization, both the average daily minimum and maximum temperatures have experienced an upward trend. Temperature change here is measured in degrees Celsius per year. Initially, in 1891, the average minimum temperature was 5.8°C and the maximum was 15.1°C. Over the next 77 years, until 1968, these temperatures increased linearly at different rates.

For the minimum temperature, the increase was 0.019°C per year and the maximum rose by 0.011°C per year. These rates show how incremental yearly temperature shifts can add up to significant changes over time, impacting various aspects of life in the city.

Urbanization Impact

Urbanization has a profound impact on the local climate, often leading to noticeable temperature changes. As cities expand, concrete structures, reduced vegetation, and increased vehicle emissions contribute to what is known as the urban heat island effect. This results in higher temperatures compared to rural areas.

For Paris, the temperature rise documented since 1891 highlights urbanization's effect on the environment. Increased temperatures can exacerbate climate-related issues, such as air quality deterioration and increased demand for energy due to cooling needs. These changes emphasize the need for sustainable urban planning to mitigate the negative impacts of urbanization.

Rate of Change

The rate of change in temperatures provides valuable insights into the dynamics of climate change over periods. In our Paris case study, the rate of temperature increase was constant for both minimum and maximum temperatures. This kind of linear relationship is often expressed using a simple equation, reflecting the yearly change.

Given the starting points of 5.8°C and 15.1°C for minimum and maximum temperatures respectively, the equations used are:

• Minimum Temperature: \(T_{min}(x) = 5.8 + 0.019x\)
• Maximum Temperature: \(T_{max}(x) = 15.1 + 0.011x\)

These equations help predict future temperatures based on past trends, assuming conditions remain similar. Understanding these rates can assist in planning and responding to ongoing climate shifts.

Future Predictions

Future predictions about temperature changes rely heavily on analyzing past trends and conditions. In our exercise, the focus was on finding when the difference between the minimum and maximum temperatures in Paris would be exactly 9°C. Using linear equations, we determined that this condition occurred around 1928.5, highlighting how mathematical modeling helps predict future climatic conditions.

This prediction process is crucial for preparing for future impacts on agriculture, infrastructure, and ecosystems. By predicting future scenarios, policymakers can develop strategies to address potential issues stemming from temperature changes, ensuring more resilient urban environments.