Problem 14 - Calculus Concepts

Question

Ice Cream Sales The table lists average monthly sales for an ice cream company. Ice Cream Sales \begin{tabular}{|c|c|c|c|} \hline Month & Monthly Sales (thou. dollars) & Month & Monthly Sales (thou. dollars) \\ \hline Jan & 50 & July & 167 \\ \hline Feb & 60 & Aug & 159 \\ \hline Mar & 77 & Sept & 108 \\ \hline Apr & 96 & Oct & 75 \\ \hline May & 137 & Nov & 61 \\ \hline June & 158 & Dec & 54 \\ \hline \end{tabular} a. Write a sine model for the ice cream data. b. Use the model to estimate the average rate of change in monthly sales between September and November. c. Numerically estimate to the nearest thousand dollars the rate of change of monthly sales in October. d. Write a sentence of interpretation for the answer to parts \(b\) and \(c\)

Step-by-Step Solution

Answer

The sine model is \( f(x) = 58.5 \sin\left(\frac{\pi}{6}(x - 6)\right) + 108.5 \). Between September and November, the average monthly sales decreased by $23,500. In October, the sales were decreasing at approximately the same rate.

Step 1: Understand the Sine Model

To model the monthly sales using a sine function, we generally write it as \( f(x) = a \sin(b(x - c)) + d \). Here, \( a \) represents the amplitude, \( b \) is related to the period, \( c \) is the horizontal shift, and \( d \) is the vertical shift.

Step 2: Determine Parameters for Sine Model

1. **Amplitude \( a \)**: The amplitude is half the difference between the maximum and minimum sales values. Here, the maximum is 167 and the minimum is 50.\[ a = \frac{167 - 50}{2} = 58.5 \]2. **Vertical Shift \( d \)**: The vertical shift is the average of the maximum and minimum sales values.\[ d = \frac{167 + 50}{2} = 108.5 \]3. **Period \( b \)**: The period of a sine function is 12 months, as sales data is cyclic annually. So,\[ b = \frac{2\pi}{12} = \frac{\pi}{6} \]4. **Horizontal Shift \( c \)**: Maximum sales occurs at July (x = 7). We set \( \sin(b(x-c)) = 1 \) for maximum, giving us \( c = 6 \).

Step 3: Write the Sine Model Function

Insert the parameters into the sine model equation:\[ f(x) = 58.5 \sin\left(\frac{\pi}{6}(x - 6)\right) + 108.5 \]

Step 4: Average Rate of Change (September to November)

To find the average rate of change from September (x = 9) to November (x = 11), use:\[ \text{Average Rate of Change} = \frac{f(11) - f(9)}{11 - 9} \]Calculate \( f(9) \) and \( f(11) \) using the model. \[ f(9) = 58.5 \sin\left(\frac{\pi}{6}(9 - 6)\right) + 108.5 \approx 108 \] \[ f(11) = 58.5 \sin\left(\frac{\pi}{6}(11 - 6)\right) + 108.5 \approx 61 \]Thus, \[ \text{Average Rate of Change} = \frac{61 - 108}{2} = -23.5 \] thousand dollars per month.

Step 5: Estimated Rate of Change at October

Numerically estimate the rate of change for October (x = 10) using a central difference method:\[ \text{Rate of Change at October} \approx \frac{f(11) - f(9)}{11 - 9} = -23.5 \] thousand dollars per month.

Step 6: Interpretation of Results

Between September and November, the average sales dropped by approximately $23,500 per month. At October, sales were decreasing at about the same rate of $23,500 per month, indicating a steady decline during this period.

Key Concepts

Amplitude

In the context of a sine model that represents cyclical data, the amplitude is an important characteristic. It represents half the difference between the peak (maximum) and bottom (minimum) values of the data set. Think of it as a measure of how "tall" or "short" the wave appears. In our ice cream sales example, the amplitude provides the swing in sales volume over the year.
To calculate the amplitude, take the maximum sales minus the minimum sales and divide by two. For instance, given our data, the maximum sales are 167,000 dollars, and the minimum is 50,000 dollars:

• Maximum sales: 167
• Minimum sales: 50

Thus, the amplitude \[a = \frac{167 - 50}{2} = 58.5 \text{~thousand dollars}\] This value indicates how much sales typically vary from the center throughout the year.

Vertical Shift

The vertical shift in a sine model represents how far the entire wave is moved up or down on the graph. This shift indicates the mean position of the wave compared to the horizontal axis. In our ice cream sales context, the vertical shift tells us the average sales level around which monthly sales fluctuate.
The formula for the vertical shift is the average of the maximum and minimum values:

• Maximum sales: 167
• Minimum sales: 50

We calculate the vertical shift as:\[d = \frac{167 + 50}{2} = 108.5 \text{~thousand dollars}\]This means that throughout the year, sales average around 108.5 thousand dollars per month, with fluctuations occurring above and below this level.

Period of a Sine Function

The period of a sine function is the length over which the wave pattern repeats itself. For annual data like ice cream sales across months, the period is typically one year or 12 months. The period helps us understand how often the pattern of sales high and low points repeat each year.
To find the coefficient \(b\) in the sine function, we relate it to the period with the formula:\[b = \frac{2\pi}{\text{period}}\]For our ice cream sales model:

• Period: 12 months

\[b = \frac{2\pi}{12} = \frac{\pi}{6}\]This value of \(b\) reflects how the monthly cycles are distributed over a year, capturing seasonal variations in sales such as summer peaks and winter troughs.

Average Rate of Change

The average rate of change offers an understanding of how a particular quantity changes over a given interval. Specifically, it represents the change in the value over a set distance, here interpreted as time. For ice cream sales data, the average rate of change tells us how fast sales increase or decrease within a specified period.
In the example, the average rate of change between September (\(x=9\)) and November (\(x=11\)) is calculated using:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]Where \(f(x)\) is the sales at month \(x\). Here:

• \(f(9) \approx 108\)
• \(f(11) \approx 61\)
• \(x_2 - x_1 = 11 - 9 = 2\)

Thus,\[\text{Average Rate of Change} = \frac{61 - 108}{2} = -23.5 \text{~thousand dollars per month}\]This negative value indicates a moderate decrease in sales of approximately $23,500 per month between September and November, highlighting the decline typically seen during the transition into fall and winter months.