Question

The monthly normal temperature \(T\) (in degrees Fahrenheit) for Pittsburgh, Pennsylvania can be modeled by \(T=\frac{22.329-0.7 t+0.029 t^{2}}{1-0.203 t+0.014 t^{2}}, \quad 1 \leq t \leq 12\) where \(t\) is the month, with \(t=1\) corresponding to January. Use a graphing utility to graph the model and find all absolute extrema. Interpret the meaning of these values in the context of the problem.

Short answer

The absolute maximum temperature corresponds to the month with the highest average temperature, and the absolute minimum temperature corresponds to the month with the lowest average temperature. The actual temperatures and months would need to be calculated using a graphing utility as specified in the step-by-step solution.

Step-by-step solution

Understanding the model

The first step involves understanding the given model. The equation provided is a rational function of \(t\), where \(t\) is the time in months starting from January. The function \(T = \frac{22.329 - 0.7t + 0.029t^2}{1 - 0.203t + 0.014t^2}\) represents the average temperature \(T\) in Pittsburgh for a given month \(t\). This function can be graphed using a graphing utility. The range \(1 \leq t \leq 12\) corresponds to the months of January through December.

Graphing the Function and Finding Extrema

Graphing this function over the interval \([1, 12]\), with \(t\) on the horizontal axis and \(T\) on the vertical axis, will allow us to visualize the average temperatures over the course of a year. With the graphing utility, it's possible to locate the highest and lowest points on the curve, which represent the warmest and coolest average temperatures respectively. Those are the points where the derivative of the function is equal to zero or the endpoints of the interval where the function is not defined elsewhere.

Interpret the Results

After finding the absolute maximum and minimum points, we can interpret these values in the context of the problem. The \(t\) value at the maximum point gives the month with the highest average temperature and the \(T\) value gives that temperature. Similarly, the \(t\) value at the minimum gives the month with the lowest average temperature, and the \(T\) value gives that temperature.

Key concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest points on a graph within a given interval. They help us identify the maximum and minimum values that a function reaches. In the context of a monthly temperature model, finding these extrema allows us to know which month has the highest and lowest average temperatures.

To find absolute extrema:

• Determine where the derivative of the function equals zero. These points are candidates for maximum or minimum values.
• Check the function's value at these points and the interval's endpoints.
• Compare these values to identify the absolute maximum and minimum.

Knowing the absolute extrema provides insights into seasonal temperature changes, allowing predictions about the hottest and coldest months.

Graphing Utilities

Graphing utilities are tools that help visualize mathematical functions. They are essential for understanding complex equations like rational functions, which can be difficult to interpret through algebra alone.

Using a graphing utility:

• Input the function equation.
• Set the range of interest, in this case, from January to December (\(1 \leq t \leq 12\)).
• Analyze the graph to find key features, like extrema, intercepts, and slopes.

Graphing utilities make it easier to see how the temperature changes over time, showing peaks and troughs that correspond to seasons, thus supporting further analysis and conclusions.

Monthly Temperature Model

A monthly temperature model like the one given offers a mathematical representation of average temperatures over a year. The function provided is a rational function, indicating that it combines polynomial equations in the numerator and denominator.

This model allows:

• Predictions about weather trends throughout the year.
• Understanding the impact of certain months on annual temperature patterns.
• Conclusions about seasonal shifts and typical weather conditions.

This monthly model enables us to make informed decisions about climate conditions and plan activities accordingly, while also offering insights into historical weather trends.