Problem 30 - Applied Calculus

Question

The desert temperature, \(H\), oscillates daily between \(40^{\circ} \mathrm{F}\) at \(5 \mathrm{am}\) and \(80^{\circ} \mathrm{F}\) at \(5 \mathrm{pm}\). Write a possible formula for \(H\) in terms of \(t\), measured in hours from 5 am.

Step-by-Step Solution

Answer

The formula is \(H(t) = -20 \cos\left(\frac{\pi}{12} t \right) + 60\).

Step 1: Identify the amplitude

The amplitude is half the difference between the maximum and minimum temperatures. Thus, \[\text{Amplitude} = \frac{80 - 40}{2} = 20^{\circ} \mathrm{F}.\]

Step 2: Determine the midline

The midline, or average temperature, is calculated as the midpoint between minimum and maximum temperatures. It is found by \[\text{Midline} = \frac{80 + 40}{2} = 60^{\circ} \mathrm{F}.\]

Step 3: Establish the period

Since the temperature oscillates from 5 am to 5 pm, which is a 12-hour interval, the complete period of the function is 24 hours. Thus, the period \((T)\) is 24, and the angular frequency \(b\) can be calculated by \[b = \frac{2\pi}{T} = \frac{2\pi}{24} = \frac{\pi}{12}.\]

Step 4: Determine the phase shift

The phase shift is determined by noting that the minimum temperature occurs at 5 am (\(t = 0\)). Since cosine functions have a minimum at \(\pi\), we suspect a function of the form \(-\cos\). No additional phase shift is necessary.

Step 5: Write the cosine function

Now combine your findings to write the cosine function. The general form is \[H(t) = a \cos(bt - c) + d.\] Inserting the specific values: \[H(t) = -20 \cos\left(\frac{\pi}{12} t \right) + 60.\] This represents the temperature \(H\) as a function of time \(t\), measured in hours from 5 am.

Key Concepts

Amplitude

The amplitude of a trigonometric function represents the height of the peaks or the depth of the troughs from the midline, which is the center of the wave. Amplitude captures how far the function's values extend from the average temperature. For scenarios like daily temperature variations, the formula to find the amplitude is quite straightforward. To calculate the amplitude ( A ), you take half the difference between the maximum and minimum values:

• First, find the highest temperature, which in this case is 80°F.
• Then, find the lowest temperature, here it is 40°F.
• Subtract the minimum from the maximum: 80°F - 40°F = 40°F.
• Finally, divide by two: 40°F / 2 = 20°F.

This means the peak and trough vary by 20°F from the midline, illustrating how widely the temperature changes throughout the day in the desert setting. Amplitude tells us the strength of the wave's oscillation, giving insights into variability, and is essential in constructing the complete formula.

Midline

The midline of a trigonometric function is essentially the line about which the function oscillates. In real-world terms, it can be perceived as the average of maximum and minimum values within a cycle.The midline formula is a simple arithmetic mean:

• Add the maximum value to the minimum value.
• Divide by two.

In our scenario, the formula is:60°F, calculated as: \((80 + 40) / 2 = 60\).
The midline, therefore, is 60°F. This line represents the mean temperature across the day and provides the position from which to measure deviations when calculating amplitude and defining how the temperature curve will rise and fall.Understanding the midline helps us to visualize the center of the trigonometric graph and contextualize average conditions experienced over time.

Phase Shift

The phase shift in trigonometric functions accounts for horizontal translations, determining where the wave starts along the horizontal axis.In this specific context, we need to identify at what time the minimum value occurs since the function typically starts at the highest or lowest point, depending on the trigonometric function form.For sinusoidal functions like cosine or sine:

• The phase shift dictates how the wave is adjusted left or right from the usual start point in trigonometric graphs.

Here, since the minimum temperature is observed at the starting point (5 am, corresponding to \(t = 0\) in hours), no phase shift is necessary.The function can straightforwardly start at its nadir with a cosine function.Consequently, no additional adjustments are made for the phase.

Period of a Function

The period of a trigonometric function indicates the duration required for one complete cycle of the waveform to occur.For functions modeling daily shifts like temperature changes, it’s relevant to measure how the function's pattern repeats over time.
To find the period (T), look for points in the cycle where the function exhibits identical behavior:

• In this instance, temperature completes one cycle from 5 am to 5 am the next day, equivalent to 24 hours.

Calculating the frequency component (b) involves converting the period into radians.Use the formula: \[b = \frac{2\pi}{T} = \frac{2\pi}{24} = \frac{\pi}{12}\]Understanding the period is crucial for constructing the function, as it captures how temperature ranges rise and fall predictably every 24 hours, reflecting the cyclical nature of the desert's climate pattern.