Question

The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21 ). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude \(40^{\circ}\) north, the latest sunrise occurs on January 4 at 7: 25 a.m. ( 14 days after the solstice), and the earliest sunset occurs on December 7 at 4: 37 p.m. ( 14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4: 30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7: 32 p.m. ( 14 days before the solstice). Using sine functions, devise a function \(s(t)\) that gives the time of sunrise \(t\) days after January 1 and a function \(S(t)\) that gives the time of sunset \(t\) days after January \(1 .\) Assume that \(s\) and \(S\) are measured in minutes and \(s=0\) and \(S=0\) correspond to 4: 00 a.m. Graph the functions. Then graph the length of the day function \(D(t)=S(t)-s(t)\) and show that the longest and shortest days occur on the solstices.

Short answer

Short Answer: To model the time of sunrise and sunset for a latitude of \(40^{\circ}\) North, we defined sine functions for sunrise \(s(t)\) and sunset \(S(t)\), and calculated their amplitudes, periods, and horizontal and vertical shifts based on given data points. We then graphed these functions and the length of the day function \(D(t) = S(t) - s(t)\), confirming that the longest and shortest days occur at the solstices.

Step-by-step solution

Defining the sine functions for sunrise and sunset

We will define sine functions \(s(t)\) and \(S(t)\) for sunrise and sunset times, respectively, with the general form \(y(t) = A\sin(Bt+C)+D\). We have the following data points: latest sunrise on January 4 (97 minutes past 4:00 a.m. at t=3), earliest sunset on December 7 (277 minutes past 4:00 p.m. at t=-25), earliest sunrise on July 2 (-30 minutes past 4:00 a.m., at t=182), and latest sunset on June 7 (452 minutes past 4:00 p.m. at t=157). Since the time difference between the latest and earliest sunrise is 127 minutes, and the time difference between the earliest and latest sunset is 175 minutes, and the function is periodic every 365 days, we can deduce the following: 1. The sine functions should have a period of \(B = \frac{2\pi}{365}\). 2. The amplitude of \(s(t)\) is \(\frac{127}{2}\), and the amplitude of \(S(t)\) is \(\frac{175}{2}\). 3. Since the peak (latest sunrise) and trough (earliest sunrise) of the \(s(t)\) graph happen 14 days after the solstices, the horizontal shift of \(C_s\) for sunrise occurs at \(14 + 182 = 196\) days. Therefore, \(C_s = B\cdot 196\). 4. Similarly, peak (latest sunset) and trough (earliest sunset) of the \(S(t)\) graph happen 14 days before the solstices. So, the horizontal shift \(C_S = B\cdot(157 - 14)\). 5. Finally, to find the vertical shift, we consider the average of the sunrise and sunset times: For \(s(t)\), we have \(D_s = \frac{0+97}{2}\) and for \(S(t)\), we have \(D_S = \frac{452+277}{2}\). Thus, our sine functions for sunrise and sunset are: \(s(t) = \frac{127}{2}\sin( \frac{2\pi}{365}(t - 196) ) + 48.5\) \(S(t) = \frac{175}{2}\sin( \frac{2\pi}{365}(t - 143) ) + 364.5\)

Graphing the sunrise function, sunset function, and Length of the day function

Plot the functions \(s(t)\) and \(S(t)\) described above and the length of the day function \(D(t) = S(t) - s(t)\) on a graphing software such as GeoGebra, Desmos, or your favorite graphing calculator. You'll be able to see the waveform of the sunrise, sunset, and length of day functions, and observe how they closely match the given data points.

Show that the longest and shortest days are on the solstices

Inspect the graph or determine the maximum and minimum values of the length of the day function \(D(t) = S(t) - s(t)\) within the given time frame. You should see that the maximum value of \(D(t)\) occurs near June 21 (Summer solstice) and the minimum value occurs near December 21 (Winter solstice). This verifies that the longest and shortest days are indeed on the solstices.

Key concepts

Periodic Functions

Trigonometric functions like sine and cosine are incredibly useful for modeling periodic phenomena. Periodic functions repeat their values in regular intervals, which means they have a cycle or a period. For example, as in the exercise of sunrise and sunset modeling, these functions are periodic because sunrise and sunset times repeat in a cyclical manner every year.

The period of a function is the length of one complete cycle. In this case, the sine functions for sunrise and sunset have a period of 365 days because the solar cycle (apparent motion of the sun as seen from Earth) repeats annually. When creating a periodic function, it's critical to accurately determine the period to represent the phenomenon accurately using the function form:

\[ y(t) = A \sin( B t + C ) + D \]

The parameter \(B\), called the angular frequency, relates to the period by \(B = \frac{2\pi}{\text{{period}}}\). Knowing this relationship helps us understand why for a cycle that is 365 days long, we use \(B = \frac{2\pi}{365}\). Adjusting \(B\) allows us to fit the periodic nature of the data correctly.

Amplitude

Amplitude in a trigonometric function represents the height of the wave from its central axis—essentially, how "tall" the wave is. In our model, it determines the extent to which sunrise and sunset times deviate from their average time. Simply put, amplitude shows how far the sunrise or sunset time can be from the average time throughout the year.

For the function \(s(t)\) representing sunrise, the amplitude is \(\frac{127}{2}\) minutes. This represents the half-range of the earliest and latest sunrise times.

The amplitude of the \(S(t)\) function, representing sunset, is \(\frac{175}{2}\) minutes, corresponding to the variability in sunset times.

A higher amplitude in a periodic function implies a greater variability in the value throughout its period. Hence, in scenarios portraying natural occurrences like this one, paying attention to the amplitude gives us insight into the fluctuation range from the average point.

Phase Shift

Phase shift refers to the horizontal displacement of the wave from its standard position, essentially determining at what point in its cycle the function starts. This helps to align the trigonometric function with the actual data's cycle.

In this exercise, a phase shift adjusts the sine functions to align them with the actual earliest and latest sunrise and sunset occurrences after the solstice dates. The phase shift allows the sine functions to accurately reflect the timing of these natural events within the year.

The phase shift \(C\) for sunrise, \(s(t)\), is calculated using the formula \(C = B \cdot 196\). This aligns the peak (latest sunrise) to occur 14 days after the solstice.

Similarly, the phase shift \(C\) for sunsets, \(S(t)\), aligns the data to 14 days before the solstice for maximum accuracy. Thus, it uses \(C = B \cdot 143\). This adjustment is crucial as it ensures the modeled wave's crest and trough appropriately match the observed phenomenon each year.

Graphing Functions

Graphing functions like \(s(t)\) for sunrise and \(S(t)\) for sunset provides a visual representation of the cyclical nature of sunrise and sunset times throughout the year. By plotting these functions, you can observe their periodic motion and how these align with real-world occurrences.

Using various graphing tools such as GeoGebra, Desmos, or any graphing calculator allows you to visualize the waveforms. With these graphs, it's easier to check how well the function models fit the data points, showing you the trends throughout the months.

Graphs also help to explore key features like maxima and minima. For instance, in this exercise when graphing the length of the day function \(D(t) = S(t) - s(t)\), you can visibly determine where the longest and shortest days occur. The maxima, corresponding to the summer solstice, and minima, at the winter solstice, confirm the periodic nature of daylight throughout the year, reinforcing how accurately the function represents the changes in day length.