Problem 8 - College Trigonometry

Question

The table below lists the average temperature of Lake Erie as measured in Cleveland, Ohio on the first of the month for each month during the years \(1971-2000 .^{19}\) For example, \(t=3\) represents the average of the temperatures recorded for Lake Erie on every March 1 for the years 1971 through 2000 . $$ \begin{array}{|l|r|r|r|r|r|r|r|r|r|r|r|r|} \hline \text { Month } & & & & & & & & & & & & \\ \text { Number, } t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \begin{array}{l} \text { Temperature } \\ \left({ }^{\circ} \mathrm{F}\right), T \end{array} & 36 & 33 & 34 & 38 & 47 & 57 & 67 & 74 & 73 & 67 & 56 & 46 \\ \hline \end{array} $$ (a) Using the techniques discussed in Example 11.1.2, fit a sinusoid to these data. (b) Using a graphing utility, graph your model along with the data set to judge the reasonableness of the fit. (c) Use the model you found in part 8 a to predict the average temperature recorded for Lake Erie on April \(15^{\text {th }}\) and September \(15^{\text {th }}\) during the years \(1971-2000 .^{20}\) (d) Compare your results to those obtained using a graphing utility.

Step-by-Step Solution

Answer

The sinusoidal model is \( T(t) = 20.5 \, \cos\left(\frac{\pi}{6}(t - 2)\right) + 53.5 \). Predicted temperatures: April 15: \( 47.1 \, ^{\circ}F \), September 15: \( 67.4 \, ^{\circ}F \).

Step 1: Identify the Sinusoidal Model

We want to fit a sinusoidal function to the temperature data. A general form for the sinusoidal function can be expressed as \( T(t) = A \cos(B(t - C)) + D \), where \( A \) is the amplitude, \( B \) affects the period, \( C \) is the phase shift, and \( D \) is the vertical shift. We will use the given data to estimate these parameters.

Step 2: Calculate Amplitude and Vertical Shift

From the data, find the maximum and minimum temperatures. The amplitude \( A \) is half the difference between the maximum and minimum temperatures: \( A = \frac{74 - 33}{2} = 20.5 \, ^{\circ}F \). The vertical shift \( D \) is the average of the maximum and minimum temperatures: \( D = \frac{74 + 33}{2} = 53.5 \, ^{\circ}F \).

Step 3: Determine Period and Phase Shift

The temperature variation completes a full cycle within a year, corresponding to 12 months. Thus, the period \( P \) is 12, so \( B \) is \( \frac{2\pi}{P} = \frac{2\pi}{12} = \frac{\pi}{6} \). The phase shift \( C \) will align the function with the data; since the data has a low point at \( t = 2 \) (February), we can use \( C = 2 \).

Step 4: Construct the Sinusoidal Equation

Using the values calculated, we have the sinusoidal model: \[ T(t) = 20.5 \,\cos\left(\frac{\pi}{6}(t - 2)\right) + 53.5 \]

Step 5: Graph the Model and Data

Using a graphing utility, plot both the sinusoidal model and the actual temperature data. Verify that the sinusoidal graph approximately fits the data points and follows the seasonal pattern indicated by the data.

Step 6: Predict Temperatures for April 15 and September 15

To find temperatures for April 15 and September 15, substitute \( t = 4 + \frac{15}{30} = 4.5 \) and \( t = 9 + \frac{15}{30} = 9.5 \) into the sinusoidal equation:\[ T(4.5) = 20.5 \,\cos\left(\frac{\pi}{6}(4.5 - 2)\right) + 53.5 \] \[ T(9.5) = 20.5 \,\cos\left(\frac{\pi}{6}(9.5 - 2)\right) + 53.5 \]

Step 7: Compute the Temperature Values

Calculating these, we have:\[ T(4.5) \approx 47.1 \, ^{\circ}F \] \[ T(9.5) \approx 67.4 \, ^{\circ}F \]This gives the predicted temperatures for those dates using the model.

Step 8: Validate with Graphing Utility

Use the graphing utility to check the values of the sinusoidal model at \( t = 4.5 \) and \( t = 9.5 \). Ensure that the predictions closely match or reasonably approximate the positions indicated by the graph.

Key Concepts

Amplitude Calculation

In the context of sinusoidal functions, amplitude is crucial as it represents half the distance between the maximum and minimum values of the function. This is particularly relevant when trying to model real-world data, such as temperatures that vary seasonally. To determine the amplitude from a given set of temperature data, you need to identify the highest and lowest temperature readings.

For example, in our data set for Lake Erie temperatures, the maximum temperature is 74°F and the minimum is 33°F. The amplitude calculation involves finding half of the range of these temperatures. Hence, the formula for amplitude is:

• Find the maximum and minimum values in the data set.
• Subtract the minimum value from the maximum value.
• Divide this result by 2.

Applying this, you get:\( A = \frac{74 - 33}{2} = 20.5 \, ^{\circ}F \).

Thus, in this situation, the amplitude of our model is 20.5°F, indicating how much the temperature fluctuates above and below the average. This value is pivotal in crafting a sinusoidal function that mirrors the natural ebb and flow of temperatures over the year.

Phase Shift

Phase shift in a sinusoidal function determines the horizontal movement of the wave along the time axis, aligning our mathematical model more accurately with observed data. Knowing where to shift the curve allows us to synchronize it with the cyclical nature of real-world situations, like temperature changes over time.

In the example with Lake Erie's temperatures, a significant point in the data, such as a low temperature value in February (at month 2, when compared to the annual cycle), helps identify the appropriate phase shift. This is your guide:

• Recognize the seasonal point of interest in your data (for instance, the trough or the crest).
• Use this point to calculate how far to shift the function horizontally.

Since the observed low point is at t = 2 (February), a reasonable phase shift C can be set as 2 in the sinusoidal equation. Incorporating phase shift into the sinusoidal model helps in accurately predicting subsequent temperature patterns by ensuring the model starts and follows the natural peaks and troughs evident in the data.

Graphing Utility

Graphing utilities play an essential role when working with sinusoidal functions, offering a visual confirmation of how well our mathematical model fits the actual data. By graphing both the sinusoidal model and the original data points, we can immediately assess the accuracy of our theoretical model.

Here’s how such graphing utilities are useful:

• They plot sinusoidal equations with exact scaling, aligning with comparable real-world data.
• By comparing models with plotted data, they enable detection of disparities and offer possible adjustments.
• Graphing utilities can also facilitate checking specific calculated points (e.g., predicting temperatures on specific dates) against the model’s curve.

Use graphing tools to plot the sinusoidal function and the temperature data for Lake Erie, noticing that these utilities can highlight any mismatches between predicted values and actual measurements. This can prompt adjustments to your function, enhancing accuracy. Ultimately, graphing utilities act like blind spot mirrors, giving you a comprehensive understanding of the data-matching precision.

Temperature Prediction

Predicting temperatures, particularly using sinusoidal models, offers practical applications, such as understanding seasonal temperature variations. Our model enables temperature predictions by substituting specific time-values into the equation.

Consider predicting temperatures on April 15 and September 15—important dates for climate analyses. These calculations use intermediate values:

• April 15 at \( t = 4 + \frac{15}{30} = 4.5 \)
• September 15 at \( t = 9 + \frac{15}{30} = 9.5 \)

This approach not only informs potential weather outcomes but also emphasizes the usefulness of sinusoidal models. Incorporate chosen dates in your equation:\[ T(4.5) = 20.5 \, \cos\left(\frac{\pi}{6}(4.5 - 2)\right) + 53.5 \]\[ T(9.5) = 20.5 \, \cos\left(\frac{\pi}{6}(9.5 - 2)\right) + 53.5 \]Resultant values, approximately 47.1°F for April 15 and 67.4°F for September 15, provide estimated expectations based on historical data trends.

By understanding and predicting temperatures, people involved in various sectors—including agriculture, hospitality, and urban planning—can make informed decisions, improving readiness for weather conditions.