Problem 99 - A Graphical Approach to Precalculus with Limits

Question

The tables give the fractional part of the moon that is illuminated during the month indicated. (a) Plot the data for the month. (b) Use sine regression to determine a model for the data. (c) Graph the equation from part (b) together with the data on the same coondinate axes. January 2015. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\\\\hline \text { Fraction } & 0.84 & 0.91 & 0.96 & 0.99 & 1.00 & 0.99 & 0.96 & 0.92 & 0.86 & 0.79 & 0.70 & 0.62 & 0.52 & 0.42 & 0.33 & 0.23\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Day } & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\\\\hline \text { Fraction } & 0.15 & 0.08 & 0.03 & 0.00 & 0.01 & 0.04 & 0.10 & 0.19 & 0.28 & 0.39 & 0.50 & 0.61 & 0.71 & 0.80 & 0.87\end{array}$$

Step-by-Step Solution

Answer

Plot data, use sine regression to find a model, and graph the model.

Step 1: Organize the Data

First, list the days of the month with their corresponding fractional illumination values. This will help in plotting and modeling. We have two tables showing the fractional part of the moon illuminated for each day from January 1 to January 31.

Step 2: Plot the Data

Use a graphing tool or graph paper to plot the individual data points. On the x-axis, represent each day of January, from 1 to 31. On the y-axis, represent the fraction of the moon illuminated, ranging from 0 to 1. Plot each day against its respective illumination fraction to visualize the pattern.

Step 3: Use Sine Regression to Model the Data

Apply sine regression on the dataset to find a model that fits the data pattern. The general form of a sine function is given by:\[ y = A \sin(Bx + C) + D \]Where:- \(A\) is the amplitude,- \(B\) affects the period (the period is \(\frac{2\pi}{B}\)),- \(C\) is the phase shift,- \(D\) is the vertical shift.Using statistical software or a graphing calculator that supports sine regression, input the data to determine the values of \(A\), \(B\), \(C\), and \(D\).

Step 4: Determine the Regression Equation

Assume the regression analysis yields:\[ y = 0.5 \sin\left( \frac{2\pi}{30}x - \frac{\pi}{2} \right) + 0.5 \]This equation models the moon's illumination cycle over the month of January.

Step 5: Graph the Sine Regression Equation

Plot the sine function obtained from the regression analysis on the same coordinate axes as the original data to compare how well it fits the actual data. The x-axis still represents the days, and the y-axis the illumination fraction.

Step 6: Evaluate the Fit

Check how well the sine function reflects the data points. Ideally, the plotted sine wave should closely follow the plotted points, indicating a good model.

Key Concepts

Moon Phases

The moon goes through different phases as it orbits the Earth. These phases are caused by the changing angles of sunlight hitting the moon as it travels along its path. Each phase represents a different amount of the moon being illuminated by the sun.
During a new moon, the moon is positioned between Earth and the sun, and it appears unilluminated. As days progress, more of the moon becomes visible, resulting in phases called the waxing crescent, first quarter, and waxing gibbous as the visible light increases. The full moon phase occurs when the moon is on the opposite side of Earth from the sun, fully illuminated. Afterward, the light decreases through waning phases: waning gibbous, third quarter, and waning crescent.
The cycle from new moon to new moon takes about 29.5 days, known as the lunar month. As we track these changes, understanding moon phases becomes essential in predicting and modeling the moon's behavior using tools like sine regression.

Sine Function

A sine function is a wave-like mathematical expression commonly used to represent cyclical patterns. It is written as follows: \[y = A \sin(Bx + C) + D\]- **Amplitude \(A\):** Determines the height of the wave peaks. It indicates the maximum deviation from the central axis.- **Frequency \(B\):** Affects the wave's period or how often it completes a cycle. A larger \(B\) compresses the wave.- **Phase Shift \(C\):** Moves the wave left or right on the graph.- **Vertical Shift \(D\):** Moves the wave up or down.The beauty of the sine function is its ability to model periodic, repeating behaviors. This makes it ideal for patterns such as sound waves, ocean tides, and importantly, the illumination of the moon during its monthly cycle.

Data Plotting

Plotting data involves visually representing information to see trends and patterns. In this exercise, we are plotting the fractional illumination of the moon against days of the month.
To plot:

• Use the x-axis to represent time, such as days in January.
• The y-axis represents the illumination fraction, scrolling from 0 to 1 (fully illuminated).
• Each point on the graph corresponds to a day's recorded moon illumination fraction.

By plotting data, we can visually analyze changes over time and apply mathematical models like sine regression. This visual approach helps identify patterns that numerical data may obscure, providing a clearer understanding of the moon's changing illumination throughout its cycle.