Problem 73 - Algebra and Trigonometry with Analytic Geometry

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 \(6 \mathrm{~A} . \mathrm{M}\). (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

(a) Tabulate values at \(t = -6, -3, 0, 3, 6,...\. (b) Max T at 6 P.M., min at 6 A.M.; Max H at 6 A.M., min at 6 P.M. (c) Inversely related due to cosine effects.

Step 1: Convert Time to Equation Input

Since time is measured from 6 A.M. as \(t = 0\), we first establish the relationship between the hours of the day and the variable \(t\). At midnight (6 hours before 6 A.M.), \(t = -6\). Subsequent times are then calculated as \(t = 0\) at 6 A.M., \(t = 6\) at 12 P.M., \(t = 12\) at 6 P.M., and \(t = 18\) at 12 A.M. the next day.

Step 2: Tabulate Temperature and Humidity Every Three Hours

For each three-hour interval from midnight:- **At Midnight (t = -6):** - Calculate \(T(-6) = -12 \cos(-\frac{\pi}{2}) + 60 = 60\). Calculate \(H(-6) = 20 \cos(-\frac{\pi}{2}) + 60 = 60\).- **At 3 A.M. (t = -3):** - Calculate \(T(-3) = -12 \cos(-\frac{\pi}{4}) + 60\) and \(H(-3) = 20 \cos(-\frac{\pi}{4}) + 60\).- **At 6 A.M. (t = 0):** - Calculate \(T(0) = -12 \cos(0) + 60 = 48\) and \(H(0) = 20 \cos(0) + 60 = 80\).- Continue calculations similarly for \(t = 3\), \(6\), \(9\), and so on to complete the day.

Step 3: Calculate Maximum and Minimum Values

Determine the times when the functions achieve their maximum and minimum values.- The cosine function achieves its maximum value of 1 and minimum value of -1.- For \(T(t)\): Minimal temperature occurs when \(\cos(\frac{\pi}{12} t) = 1\) making \(T_{min} = 48\). Maximal temperature occurs when \(\cos(\frac{\pi}{12} t) = -1\) leading to \(T_{max} = 72\) .- For \(H(t)\): Maximum humidity occurs when \(\cos(\frac{\pi}{12} t) = 1\) giving \(H_{max} = 80\), and minimum when \(\cos(\frac{\pi}{12} t) = -1\) resulting in \(H_{min} = 40\).- Solve for \(t\) using \(\frac{\pi}{12} t = 0, \pi\) and \(2\pi,..\) to find these times throughout the day.

Step 4: Discuss Temperature and Humidity Relationship

The cosine components \(-12\cos\) and \(20\cos\) determine the variations from the mean values for temperature and humidity. As the value of cosine changes over time, temperature and humidity are inversely affected due to opposite cosine coefficients (-12 for \(T\), +20 for \(H\)). Analyze how temperature decreases while humidity increases when cosine moves from 1 to -1.

Key Concepts

Temperature Modeling

Temperature modeling is a fascinating process that involves using equations to understand and predict temperature changes over a time period. In the given exercise, the temperature in Tucson, Arizona, on March 17, 1981, is modeled with the equation: \[ T(t) = -12 \cos \left( \frac{\pi}{12} t \right) + 60. \]This equation uses the cosine function, a popular choice for its ability to represent cyclical patterns like temperature changes over a day.

• The constant "60" here represents the average temperature around which daily fluctuations occur.
• The "-12" is the amplitude, which dictates how much the temperature deviates above and below the average.
• The term \( \frac{\pi}{12} t \) inside the cosine function determines the rate of temperature changes over time or its periodicity.

Through this mathematical model, we can make accurate predictions at specific times of the day, understanding how daytime warming and nighttime cooling play out.

Relative Humidity

Relative humidity is the amount of moisture in the air in relation to what the air can hold at that temperature. On this specific day, humidity is expressed by the function: \[ H(t) = 20 \cos \left( \frac{\pi}{12} t \right) + 60. \]This function helps track how moisture levels fluctuate throughout the day, independent of temperature. This is essential for understanding weather patterns and making forecasts.

• The constant "60" here denotes the average humidity percentage for the day.
• The "20" in front of the cosine term signifies the amplitude, influencing how much humidity can deviate from its average value.
• Similar to temperature, \( \frac{\pi}{12} t \) governs the wave pattern of humidity changes.

This model can be used to predict humidity levels at different times, such as morning or late evening, essential information for activities like farming or determining comfort levels for outdoor events.

Cosine Function

The cosine function plays a central role in modeling both temperature and humidity in this problem. As a cyclical function, cosine is inherently periodic and symmetric, making it ideal for representing patterns where the values repeat over a regular interval like a day.

• The cosine curve's basic form can be described by elements such as amplitude, frequency, and period, all of which allow it to adapt and fit the desired cyclical pattern.
• In \[ T(t) = -12 \cos \left( \frac{\pi}{12} t \right) + 60 \, \text{and} \, H(t) = 20 \cos \left( \frac{\pi}{12} t \right) + 60, \] changes in the cosine term dictate the rises and falls in temperature and humidity respectively.
• Cosine values range from \(-1\) to \(1\), explaining why during a full cycle, modeled parameters experience periodic highs and lows, reaching maximum and minimum values at certain times.

Understanding the cosine function’s properties is key for analyzing and predicting cycles within natural phenomena, including but not limited to weather variables.

Wave Patterns

Wave patterns refer to the rhythmic rise and fall depicted by trigonometric functions like sine or cosine. In the context of this exercise, temperature and humidity changes are portrayed as wave patterns using a cosine function. This pattern is repetitive and consistent over time, reflecting natural cycles such as day-night temperature fluctuations.

• The amplitude of wave patterns influences how high and low the temperature or humidity spikes and dips over the cycle.
• Each wave’s troughs and crests align with specific times of day, determining when a maximum or minimum occurs.
• The wave period, derived from \( \frac{\pi}{12} t \), indicates how quickly the full cycle of change happens across a 24-hour span.

By understanding wave patterns, you can better grasp when temperatures peak (usually in daylight) and when humidity spikes (often at night), critical insights for weather analysis.

Amplitude and Periodicity

Amplitude and periodicity are key features of wave motions in trigonometric functions that greatly affect how we analyze the equations. Amplitude defines the height of the wave, reflecting the strength of change in temperature and humidity.

• The amplitude of "-12" for temperature and "20" for humidity tells us the extent of their daily fluctuation from the average.
• Periodicity involves the duration over which the complete wave cycle occurs. Here, the period is identified through the coefficient \( \frac{\pi}{12} \) which orchestrates how temperatures and humidity oscillate once every 24 hours.
• A full wave cycle of cosine demonstrates how these variables regularly increase and decrease over time, essential for forecasting purposes.

Recognizing amplitude and periodicity provides deeper insights into the environmental rhythm, allowing for precise adjustments in predictions for daily and seasonal forecasts.