Problem 54 - Precalculus: Functions and Graphs

Question

Graph the equation \(y=f(t)\) on the inter val \([0,24] .\) Let \(y\) represent the outdoor temperature \(\left(\text { in }^{\circ} \mathrm{F}\right)\) at time \(t\) (in hours), where \(t=0\) corresponds to 9 A.M. Describe the temperature during the 24 -hour interval. \(y=80+22 \cos \left[\frac{\pi}{12}(t-3)\right]\)

Step-by-Step Solution

Answer

The temperature peaks at 102°F at noon and reaches 58°F at midnight, following a sinusoidal pattern over 24 hours.

Step 1: Understand the scenario

The given function \(y = f(t) = 80 + 22 \cos\left[\frac{\pi}{12}(t-3)\right]\) models the outdoor temperature (in degrees Fahrenheit) over a 24-hour period starting at 9 A.M. Here, \(t = 0\) corresponds to 9 A.M., and \(t\) increases by 1 hour until it reaches 24 hours, representing 9 A.M. the next day.

Step 2: Identify properties of the cosine function

The function \(22 \cos\left[\frac{\pi}{12}(t-3)\right]\) is a cosine function with an amplitude of 22, a period of \(24\) hours (since the factor \(\frac{\pi}{12}\) makes it complete a full cycle over 24 hours), and a horizontal shift to the right by 3 hours. The cosine component of the equation represents the oscillation around the mean temperature.

Step 3: Determine mean temperature

The mean temperature is given by the constant term, \(80\). This is the midpoint around which the temperature oscillates due to the cosine term.

Step 4: Calculate temperature extremes

The maximum temperature is \(80 + 22 = 102^{\circ} F\) and the minimum temperature is \(80 - 22 = 58^{\circ} F\). These values occur because the cosine term varies between \(-1\) and \(+1\).

Step 5: Analyze temperature changes over time

From the properties of the cosine function and the time shift, the maximum temperature will occur at \(t=3\) (12 PM), the minimum at \(t=15\) (midnight next day), and returns to 80 at \(t=0\) (9 AM) and \(t=24\).

Step 6: Sketch the graph

Plot the function over the interval \([0, 24]\). Start with a minimum at \(t=0\), reach max at \(t=3\), return to 80 at \(t=6\), minimum at \(t=15\), and finally return to 80 at \(t=24\), showing the sinusoidal pattern of temperature changes.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It's often used to model periodic phenomena such as sound waves, light waves, and in this case, temperature changes over time. The basic form of the cosine function is given by \(y = a \cos(bx + c) + d\), where it describes how the function shifts in response to changes in amplitude, frequency, and phase shifts.

• The cosine function is useful in modeling scenarios where a regular, repeating pattern occurs.
• It completes one full cycle from maximum to minimum to maximum.
• In the temperature model, it captures daily temperature variations with a regular periodicity.

By analyzing cosine graphs, we can identify key characteristics like maximum and minimum values in this periodic pattern.

Amplitude

Amplitude refers to the maximum distance a wave travels from its average position (in this case, the midline or mean temperature). In our temperature model, the amplitude is represented by the coefficient of the cosine term, 22.

• The amplitude affects how tall the peaks of the wave are, representing the amount of variation from the average temperature.
• In the given function \(22 \cos\left[\frac{\pi}{12}(t-3)\right]\), the amplitude is 22 degrees.

This means the temperature fluctuates 22 degrees above and below the mean temperature of 80°F, ensuring we capture realistic highs and lows in daily temperature patterns.

Period

The period of a trigonometric function is the time taken to complete one full cycle of the waveform. For the cosine function, a complete cycle is from its peak back to its next peak.

• In mathematical terms, the period is calculated as \(\frac{2\pi}{b}\), where \(b\) is the frequency factor of the function.
• In our function \(22 \cos\left[\frac{\pi}{12}(t-3)\right]\), \(b\) is \(\frac{\pi}{12}\).
• This makes the period: \(\frac{2\pi}{\frac{\pi}{12}} = 24\) hours.

Thus, our function indicates that the temperature goes through its whole cycle just once every 24 hours, representing the full daily cycle, from peak warmth to coolest and back.

Horizontal Shift

Horizontal shift, or phase shift, refers to shifting the function from its usual starting point along the horizontal axis. In this exercise, the cosine function is shifted horizontally, indicating the time alignment of the temperature cycle within a day.

• The horizontal shift is determined by the term \((-3)\) inside the cosine, indicating a shift to the right by 3 hours.
• This means that rather than starting at maximum temperature at \(t = 0\) (9 AM), the cosine function reaches its peak at \(t = 3\) (12 PM).
• This shift aligns with typical temperature patterns where the temperature might peak around midday.

Understanding the horizontal shift helps in aligning the mathematical model with real-world timing, such as identifying the hottest and coldest times of the day within the model.