Problem 73 - Precalculus: Functions and Graphs

Question

On March \(17,1981,\) in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation $$T(t)=-12 \cos \left(\frac{\pi}{12} t\right)+60$$ while the relative humidity in percent could be expressed by $$H(t)=20 \cos \left(\frac{\pi}{12} t\right)+60$$ where \(t\) is in hours and \(t=0\) corresponds to 6 A.M. (a) Construct a table that lists the temperature and relative humidity every three hours, beginning at midnight. (b) Determine the times when the maximums and minimums occurred for \(T\) and \(H\) (c) Discuss the relationship between the temperature and relative humidity on this day.

Step-by-Step Solution

Answer

Temperature and humidity are inversely related, with temperature peaking 12 hours after humidity.

Step 1: Understand the functions

The temperature and humidity functions are represented by trigonometric equations. Recognize that \(T(t)\) and \(H(t)\) are periodic functions with a cosine component. The amplitude for \(T(t)\) is \(-12\) and for \(H(t)\) is \(20\). Both functions have a period of \(24\) hours since the term \(\frac{\pi}{12}\) causes them to complete one cycle in 24 units of time (hours here). The average (midline) for both functions is 60.

Step 2: Calculate values for T and H every 3 hours

Since \(t=0\) represents 6 A.M., find values from \(t=-6\) (midnight) to \(t=18\) (6 P.M. next day) at 3-hour intervals. Substitute these \(t\) values into the functions:- **Midnight (t=-6):** - \( T(-6) = -12\cos\left(\frac{\pi}{12}(-6)\right)+60 \) - \( H(-6) = 20\cos\left(\frac{\pi}{12}(-6)\right)+60 \)- **3 A.M. (t=-3):** Substituting \(t=-3\) in both functions.- **6 A.M. (t=0), 9 A.M. (t=3), 12 P.M. (t=6), ...** Until 6 P.M. next day (t=18).Calculate these values to construct the table.

Step 3: Identify maxima and minima for T and H

The cosine function reaches maximum at its start \(t=0\), then goes to minimum after half of its period \(t=12\). Apply these properties:- **Temperature (T):** The maximum occurs at \(t=12\) and minimum at \(t=0\).- **Humidity (H):** The maximum occurs at \(t=0\) and minimum at \(t=12\).Convert \(t=12\) and \(t=0\) into clock times to find when these extremas occur in a day.

Step 4: Discuss the relationship between temperature and humidity

Examine the functions: - They have identical periods and are phase-shifted by 12 hours (180 degrees in phase), meaning high temperature corresponds with low humidity and vice versa. - This phase difference suggests an inverse relation on this day: as temperature increases, relative humidity decreases, showing a classic temperature-humidity relationship.

Key Concepts

Temperature Modeling

Temperature modeling is crucial in understanding daily or seasonal changes in weather conditions. To model temperature, we often use functions that describe how temperature changes over time. On March 17, 1981, in Tucson, Arizona, the temperature overseen through the function \[ T(t) = -12 \cos \left( \frac{\pi}{12} t \right) + 60 \] captures the oscillating nature of the daily temperature. This equation is a periodic one that uses trigonometric functions to simulate natural variability. Such modeling helps in anticipating temperatures for each hour throughout the day. Coupled with other data, it allows meteorologists to better predict and understand climate patterns and daily weather fluctuations.

Relative Humidity

Relative humidity measures the amount of moisture in the air relative to the maximum it can hold at a specific temperature. This value is represented by the function \[ H(t) = 20 \cos \left(\frac{\pi}{12} t \right) + 60 \] for this specific day in Tucson. This equation demonstrates how relative humidity changes periodically throughout the day. The midline value of 60 indicates an average humidity level around which actual levels fluctuate. By understanding these periodic changes, forecasters can determine not only when humidity is higher or lower, but also predict potential impacts on weather conditions and comfort levels for individuals in that area.

Periodic Functions

Periodic functions describe phenomena that repeat at regular intervals, like temperature and humidity. In the world of mathematics, these functions are commonly expressed using trigonometric functions such as sine or cosine. A key characteristic of a periodic function is its period, the interval at which it repeats. In our exercise:

• Both temperature and humidity functions have a period of 24 hours.
• This implies that every 24 hours, the pattern of the temperature and humidity repeats itself entirely.

By plotting these functions, one can visualize cycles of highs and lows which indicate factors like daily routines and natural processes affecting these conditions. This makes periodic functions indispensable tools in predicting and making sense of naturally occurring patterns.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, besides sine and tangent. It is often used to model periodic phenomena because of its smooth, repetitive wave-like nature. In the temperature and humidity equations, the cosine function provides a smooth oscillation between extrema, making it a perfect choice to model daily changes. Some properties of the cosine function that are useful in this context include:

• The maximum value of a cosine function, when unaltered, is 1, and the minimum is -1.
• This makes it easy to determine amplitude and phase adjustments needed to fit real-world patterns.

In essence, cosine assists in creating a timely and regular visual of natural phenomena's behavior across time.

Amplitude and Phase Shift

Amplitude and phase shift are two critical components when working with trigonometric functions such as cosine. The amplitude measures the height of the wave from the midline of the function to its peak or trough.

• Here, the amplitude for temperature is 12, indicating how much the temperature can fluctuate above or below the average value of 60.
• For humidity, the amplitude of 20 shows similar fluctuation behavior.

Phase shift is how far the function is shifted horizontally along the time axis. For this model:

• The functions for temperature and humidity experience opposite phase shifts, explaining the time difference in peak occurrences.

Understanding these components enables accurate predictions and adjustments necessary for a well-aligned representation of environmental changes.