Question

The following data represent the average monthly temperatures for Baltimore, Maryland. (a) Draw a scatter plot of the data for one period. (b) Find a sinusoidal function of the form \(y=A \sin (\omega x-\phi)+B\) that models the data.\ (c) Draw the sinusoidal function found in part (b) on the scatter plot. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on a scatter plot of the data. $$ \begin{array}{lc} \hline & \text { Average Monthly } \\ \text { Month, } \boldsymbol{x} & \text { Temperature, }^{\circ} \mathrm{F} \\\ \hline \text { January, } 1 & 32.9 \\ \text { February, } 2 & 35.8 \\ \text { March, } 3 & 43.6 \\ \text { April, } 4 & 53.7 \\ \text { May, } 5 & 62.9 \\ \text { June, } 6 & 72.4 \\ \text { July, } 7 & 77.0 \\ \text { August, } 8 & 75.1 \\ \text { September, } 9 & 67.8 \\ \text { October, } 10 & 56.1 \\ \text { November, } 11 & 46.5 \\ \text { December, } 12 & 36.7 \end{array} $$

Short answer

The sinusoidal function modeling the data is approximately \( y = 22.05 \sin\left( \frac{\pi x}{6} - \phi \right) + 54.95 \). Use graphing tools to refine this function for the best fit.

Step-by-step solution

Create a Scatter Plot

Plot the data points from the table on a graph where the x-axis represents the months (1 to 12) and the y-axis represents the average monthly temperatures in °F for Baltimore.

Determine the Parameters

Identify the parameters for the sinusoidal function \(y = A \sin( \omega x - \phi) + B\).Find the amplitude (A), vertical shift (B), period (T), and horizontal shift (\( \phi \)), using the temperature data.Amplitude, \( A \) : Half the difference between the maximum and minimum temperatures.Vertical shift, \( B \) : Average of the maximum and minimum temperatures.Period, \( T \) : The number of months it takes for the temperature pattern to repeat (which is 12 months).Horizontal shift, \( \phi \) : It will be based on the fitting sinusoidal curve through specific range of the month.

Compute the Values

Calculate the amplitude using the formula:\[ A = \frac{77.0 - 32.9}{2} = 22.05 \]Calculate the vertical shift using the formula:\[ B = \frac{77.0 + 32.9}{2} = 54.95 \]The period for a yearly cycle is 12 months, so the angular frequency (\(\omega\)) is calculated using the formula:\[ \omega = \frac{2\pi}{12} = \frac{\pi}{6} \]Estimate the phase shift (\(\phi\)) which ensures the function best fits the data using vertical shift range or fitting single sinusoidal curve through few few period of the month steps ensuring vertical shift

Write the Sinusoidal Function

Combine the values and write down the sinusoidal function:\[ y = 22.05 \sin\left( \frac{\pi x}{6} - \phi \right) + 54.95 \]

Overlay the Sinusoidal Function on the Scatter Plot

Plot the sinusoidal function obtained in step 4 on the same graph as the scatter plot to visually verify the model.

Use a Graphing Utility for Best Fit

Utilize graphing software or a graphing calculator to input the scatter plot data and let the tool find the parameters for the sinusoidal function that best fits the data.You'll get a refined amplitude, vertical shift, period, and phase shift.

Graph the Best Fit Sinusoidal Function

Plot this refined sinusoidal function from the graphing utility on the same scatter plot to compare how closely it matches the data points.

Key concepts

Scatter Plot

To begin analyzing the average monthly temperatures in Baltimore, the first step involves creating a scatter plot. A scatter plot is a type of graph that is used to display and compare two variables. For this exercise:

• The x-axis represents the months of the year, from January (1) to December (12).
• The y-axis represents the average temperature in degrees Fahrenheit.

Each data point on the scatter plot corresponds to the temperature for a specific month. For instance, for January, the temperature is 32.9°F. Plot each of the given temperature values this way. This visual representation makes it easier to see trends or patterns in the temperature data over time.

Sinusoidal Model

A sinusoidal model is used to describe periodic data, which repeats at regular intervals. In the context of temperature data, the temperatures follow a yearly cycle, making a sinusoidal model appropriate. A sinusoidal function has the general form:



where:

• A is the amplitude
• ω (omega) is the angular frequency
• φ (phi) is the phase shift
• B is the vertical shift

This model helps in predicting temperatures for the months not given and understanding the overall pattern. By identifying the correct parameters for A, ω, φ, and B, we can construct a function that accurately represents the temperature data.

Amplitude

Amplitude is a key parameter in the sinusoidal model. It measures the height of the wave, or the difference between the midline and the peak (or trough). To calculate the amplitude (A) for the temperature data in Baltimore, you use the formula:



For our data:

• The maximum temperature (maxy ) is 77.0°F (July)
• The minimum temperature (miny) is 32.9°F (January)

Using the formula, we find:

= 22.05

The amplitude tells us how much the temperature deviates from the average value over the year.

Phase Shift

Phase shift (φ) is another important element in a sinusoidal function. It represents the horizontal shift of the wave along the x-axis. For temperature data, it indicates how the wave is displaced in relation to the time (months).

Estimate the phase shift by aligning the sinusoidal function's peak with the month that has the highest temperature. If July (month 7) has the highest value, and the theoretical peak of should occur at , then:

Period

The period (T) of a sinusoidal function is the length of one complete cycle. In the context of average monthly temperature data, the period corresponds to one year or 12 months.

Calculate the angular frequency (ω), which relates to the period, using:

(rounded)

The angular frequency helps to determine how many cycles the sinusoidal function goes through in a unit interval. For our yearly temperature data, knowing that the cycle completes in 12 months simplifies the determination of ω.

Period

The period (T) of a sinusoidal function is the length of one complete cycle. In the context of average monthly temperature data, the period corresponds to one year or 12 months.

Calculate the angular frequency (ω), which relates to the period, using:

(rounded)

The angular frequency helps to determine how many cycles the sinusoidal function goes through in a unit interval. For our yearly temperature data, knowing that the cycle completes in 12 months simplifies the determination of ω.