Problem 30 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The temperature formula is \(H(t) = 20 \sin\left(\frac{\pi}{12} t - \frac{\pi}{2}\right) + 60\).

Step 1: Identify the Problem

We need to find a formula for the temperature, \(H\), as it oscillates between \(40^{\circ} \mathrm{F}\) at 5 am and \(80^{\circ} \mathrm{F}\) at 5 pm. The variable \(t\) represents the time in hours from 5 am.

Step 2: Analyze the Temperature Oscillation

The temperature oscillates sinusoidally between 40°F (minimum) at 5 am and 80°F (maximum) at 5 pm. This indicates a periodic function can describe the temperature changes.

Step 3: Determine Amplitude, Midline, and Period

The amplitude \(A\) is half the difference between the maximum and minimum temperatures: \(A = \frac{80 - 40}{2} = 20\). The midline \(C\) is the average of the maximum and minimum temperatures: \(C = \frac{80 + 40}{2} = 60\). The period is 24 hours, since the cycle completes every day.

Step 4: Choose the Correct Sinusoidal Model

Since the oscillation is sinusoidal, we'll use the cosine function: \(H(t) = A \cos(\omega t + \phi) + C\). Cosine is chosen because it starts from the maximum.

Step 5: Calculate the Angular Frequency and Phase Shift

The angular frequency \(\omega\) is calculated as \(\omega = \frac{2\pi}{\text{Period}} = \frac{2\pi}{24} = \frac{\pi}{12}\). At \(t=0\), the temperature is at the minimum, so we use the sine formula and adjust for phase: \(H(t) = 20 \sin\left(\frac{\pi}{12} t - \frac{\pi}{2}\right) + 60\).

Step 6: Write the Final Formula

The formula for the temperature in terms of hours from 5 am is \(H(t) = 20 \sin\left(\frac{\pi}{12} t - \frac{\pi}{2}\right) + 60\). This represents a sinusoidal function that reflects the daily oscillation.

Key Concepts

Amplitude

In the context of sinusoidal functions, the amplitude represents the maximum deviation from the midline of the function. Specifically, it shows how far the peaks and valleys of the wave stretch above and below this central line. In our exercise, the amplitude is crucial because it defines how much the temperature varies throughout the day.

To calculate the amplitude, we take half of the difference between the maximum and minimum temperatures:

• The maximum temperature is 80°F.
• The minimum temperature is 40°F.

Thus, the amplitude, denoted as \(A\), is calculated as:\[A = \frac{80 - 40}{2} = 20\]This tells us that the temperature deviates 20°F above and 20°F below the average temperature during the day.

Midline

The midline of a sinusoidal function is the horizontal line that runs through the center of the wave. It represents the average value around which the function oscillates. Understanding the midline is important because it marks the equilibrium point of the function.

For the desert temperature problem, we find the midline by calculating the average of the maximum and minimum temperatures. Here's how it's done:

• Maximum temperature: 80°F
• Minimum temperature: 40°F

The midline, denoted as \(C\), is given by:\[C = \frac{80 + 40}{2} = 60\]So, the temperature oscillates around 60°F, making this the central line in the sinusoidal model.

Periodicity

Periodicity refers to the repeating nature of sinusoidal functions. The period is the length of one complete cycle of the wave before it starts to repeat itself. In the study of sinusoidal functions, understanding the period helps determine the frequency of oscillations.

In our temperature scenario, this is fairly straightforward. The pattern of temperature variation repeats every 24 hours. Therefore, the period \(T\) of the function is 24 hours. Knowing the period allows us to calculate the angular frequency \(\omega\):\[\omega = \frac{2\pi}{T} = \frac{2\pi}{24} = \frac{\pi}{12}\]This angular frequency \(\omega\) is used in the cosine or sine function to model the oscillating temperature.

Phase Shift

Phase shift is a critical feature of sinusoidal functions that indicates how much the function is horizontally shifted from a standard position. It determines where on the cycle the function starts, which is crucial for fitting sinusoidal models to real-world data.

For the desert temperature, we initially determine that the function starts with a sine wave, given that the temperature is at a minimum at 5 am. To correctly align the function with the observed temperature data, we introduce a phase shift.The formula provided, \(H(t) = 20 \sin\left(\frac{\pi}{12} t - \frac{\pi}{2}\right) + 60\), uses a phase shift of \(-\frac{\pi}{2}\). This adjustment reflects the fact that the sine wave would naturally start at zero, but here we need it to start at a minimum (40°F) at \(t=0\).

The phase shift ensures that the sine function aligns perfectly with the actual temperature readings throughout the day.