Problem 61 - Precalculus: Functions and Graphs

Question

The number of daylight hours \(D\) at a particular location varies with both the month and the latitude. The table lists the number of daylight hours on the first day of each month at \(60^{\circ} \mathrm{N}\) latitude. $$\begin{array}{|lc|c|c|cc|} \hline \text { Month } & \boldsymbol{D} & \text { Month } & \boldsymbol{D} & \text { Month } & \boldsymbol{D} \\ \hline \text { Jan. } & 6.03 & \text { May } & 15.97 & \text { Sept. } & 14.18 \\ \hline \text { Feb. } & 7.97 & \text { June } & 18.28 & \text { Oct. } & 11.50 \\ \hline \text { March } & 10.43 & \text { July } & 18.72 & \text { Nov. } & 8.73 \\ \hline \text { April } & 13.27 & \text { Aug. } & 16.88 & \text { Dec. } & 5.88 \\ \hline \end{array}$$ A. Let \(t\) be time in months, with \(t=1\) corresponding to January, \(t=2\) to February, \(\ldots, t=12\) to December, \(t=13\) to January, and so on. Plot the data for a twoyear period. B. Find a function \(D(t)=a \sin (b t+c)+d\) that ap proximates the number of daylight hours. Graph the function \(D\) with the data.

Step-by-Step Solution

Answer

Plot the data over two years, calculate sine function: \(D(t) = 6.42 \sin\left(\frac{\pi}{6}t + c\right) + 12.30\), adjust \(c\) for best fit.

Step 1: Convert Data for Two-Year Period

For each month from January to December, assign values for a two-year period with the months and respective days. Extend the months to include from 1 to 24, by repeating values for each month from the given data. This will simulate two years' worth of daylight hours variation.

Step 2: Plot the Data

Using the values from the table, plot the number of daylight hours on the Y-axis against the months on the X-axis, ranging from 1 to 24. This helps visualize the data pattern over two years.

Step 3: Identify the Sinusoidal Features

Observe the plotted data; it shows sinusoidal behavior with daylight hours peaking and dipping in a predictable pattern. Approximate the maximum value (around 18.72 hours in July) and the minimum value (around 5.88 hours in December). These will help in calculating parameters for the sinusoidal function.

Step 4: Calculate the Sinusoidal Parameters

Use the features from the plot:- **Amplitude** \(a\): Calculate as \((\text{Max} - \text{Min})/2 = (18.72 - 5.88)/2 = 6.42\).- **Vertical Shift** \(d\): Calculate as \((\text{Max} + \text{Min})/2 = (18.72 + 5.88)/2 = 12.30\).- **Period** \(b\): Since the period is 12 months, set \(b = \frac{2\pi}{12} = \frac{\pi}{6}\).- **Phase Shift** \(c\): Shift the sinusoidal curve horizontally so that it starts from a reasonable point such as March (3) when day length increases, finding \(c\) will require adjusting so the formula fits the data (typically involves solving for \(c\) when period starts increasing notably).

Step 5: Formulate the Sinusoidal Function

Using the parameters calculated, the function is: \[D(t) = 6.42 \sin\left(\frac{\pi}{6}t + c\right) + 12.30\]. Determine \(c\) through trial and error against the plotted data or via regression techniques.

Step 6: Graph the Sinusoidal Function

Now graph \(D(t) = 6.42 \sin\left(\frac{\pi}{6}t + c\right) + 12.30\) on the same plot as the data to compare and refine adjustments if necessary. Fine-tune the value of \(c\) until the function aligns closely with the peaks and troughs of the actual data.

Key Concepts

Daylight Hours

Daylight hours refer to the amount of time during the day when sunlight is visible. These hours vary depending on the time of year and your location on Earth—one significant factor being latitude. In the exercise given, we're particularly interested in a location at a latitude of \(60^{\circ} \mathrm{N}\). At such high latitudes, the variation in daylight hours is more pronounced throughout the year. For instance, during summer months, days can be incredibly long, while in winter, daylight hours are considerably short.
The table in the exercise lists the daylight hours for each month, starting from January. Understanding daylight hours allows us to make predictions and observe natural phenomena like the Midnight Sun, which occurs in places far north or south of the Equator.

Latitude Variation

Latitude plays a critical role in the variation of daylight hours. The higher the latitude, the greater the variation you will observe across the year. At the Equator, daylight hours remain nearly constant around 12 hours throughout the year. However, as you move toward the poles, variations become significant.

• At \(60^{\circ} \mathrm{N}\), as in the exercise's example, summer days can reach up to 18.72 hours of daylight in July.
• Conversely, winter days can shrink to as little as 5.88 hours in December.

The tilt of the Earth's axis causes these changes as it orbits around the sun. This axial tilt results in different portions of the Earth receiving varying levels of sunlight at different times of the year.

Data Visualization

Data visualization is a powerful tool for understanding patterns in data, like the fluctuating daylight hours over a year. By plotting the daylight hours on a graph, we gain insights into their patterns over time. This involves:

• Assigning the months as the X-axis ranging from 1 to 24 to represent a two-year period.
• Plotting the actual daylight hours from the table onto the Y-axis.

Visualizing this data helps identify the sinusoidal pattern, as you can literally see the rise and fall in daylight hours. It's similar to the wave-like behavior of a sine function, underscoring the relationship between the natural cycles and mathematical models. Such visual aids simplify understanding and pave the way for further mathematical analysis to create models like the sinusoidal function.

Sinusoidal Parameters

Defining sinusoidal parameters is essential for creating a mathematical model that mimics real-world patterns—specifically, the pattern of daylight hours over time. These parameters include:

• Amplitude (\(a\)): This represents half of the difference between the maximum and minimum daylight hours. In the exercise, it is calculated as \((18.72 - 5.88)/2 = 6.42\).
• Vertical Shift (\(d\)): This is the average of the maximum and minimum values, positioned at \((18.72 + 5.88)/2 = 12.30\).
• Period (\(b\)): The period of the cycle is 12 months, thus \(b = \frac{2\pi}{12} = \frac{\pi}{6}\).
• Phase Shift (\(c\)): The horizontal shift applied to align the function with the real data, adjustable to fit the wave starting typically where day length begins to increase, such as March.

By calculating these parameters, we can formulate a sinusoidal function, which in this scenario is written as \[D(t) = 6.42 \sin\left(\frac{\pi}{6}t + c\right) + 12.30\]. Adjusting the phase shift \(c\) through additional graph fitting ensures that the function accurately reflects the observed daylight hour data.