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Chapter 1: Problem 21

Canned soups are consumed more during colder months and less during warmer months. A soup company estimates its sales of \(16-\mathrm{oz}\) cans of condensed soup to be at a maximum of 215 million during the 5 th week of the year and at a minimum of 100 million during the 3lst week of the year. a. Calculate the period and horizontal shift of the soup sales cycle. Use these values to calculate the parameters \(b\) and for a model of the form \(f(x)=a \sin (b x+c)+d\). b. Calculate the amplitude and average value of the soup sales cycle. c. Use the constants from parts \(a\) and \(b\) to construct a sine model for weekly soup sales.

Short Answer

Expert verified
The sine model for weekly soup sales is \( f(x) = 57.5 \sin\left(\frac{\pi}{26}x - 8\right) + 157.5 \).

Step by step solution

01

Analyze the problem and calculate the period

The soup sales cycle fluctuates annually with seasons, thus repeating approximately every year (52 weeks). Therefore, the period \( T \) is 52 weeks for the function \( f(x) \). For a sine function \( f(x) = a \sin(bx + c) + d \), the period is given by \( T = \frac{2\pi}{b} \). Thus, \( b = \frac{2\pi}{T} = \frac{2\pi}{52} \approx \frac{\pi}{26} \).
02

Determine the horizontal shift

The maximum sales occur in the 5th week, suggesting that the peak of the sine wave aligns here. Sine waves typically reach their first peak at \( x = \frac{T}{4} = \frac{52}{4} = 13 \). Therefore, the horizontal shift \( c \) can be calculated to align the maximum at week 5: solve \( \frac{b(5 + c)}{b} = \pi/2 \). Substituting \( b = \pi/26 \), we get \( 5 + c = 13 \), so \( c = 8 \). Thus, \( f(x) = a \sin(\frac{\pi}{26}x - 8) + d \).
03

Calculate the amplitude and average value

The amplitude \( a \) is half the difference between maximum and minimum sales, which is \( \frac{215 - 100}{2} = 57.5 \). The average value, \( d \), is the midpoint of the maximum and minimum sales values, which is \( \frac{215 + 100}{2} = 157.5 \).
04

Construct the sine model

Substitute calculated values into the sine function: \( a = 57.5 \), \( b = \frac{\pi}{26} \), \( c = 8 \), \( d = 157.5 \). The model for weekly soup sales is \( f(x) = 57.5 \sin\left(\frac{\pi}{26}x - 8\right) + 157.5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
The amplitude of a sine wave is a crucial characteristic that measures the wave's height. It is defined as the maximum displacement of the wave from its average value.
For the soup sales cycle, the amplitude is calculated as half the difference between the maximum and minimum sales figures.
Specifically, it is the distance between the highest sales number of 215 million and the lowest number of 100 million. This is summed up as:
  • Amplitude, \( a = \frac{215 - 100}{2} = 57.5 \)
This means that the sales values can deviate up to 57.5 million above or below the average sales value during the soup cycle.
Understanding amplitude helps you anticipate the range of sales fluctuations and prepare accordingly.
Period
The period of a sine wave is the length of time it takes for the wave to complete one full cycle. For the soup sales modeled as a sine wave, the period represents how often the sales cycle repeats throughout the year.
Since soup sales rise and fall with the seasons, they complete a full cycle every 52 weeks, or one year. Mathematically, this is expressed as:
  • Period, \( T = 52 \) weeks
For a sine function expressed as \( f(x) = a \sin(bx + c) + d \), the relationship between the period and \( b \) is given by \( T = \frac{2\pi}{b} \).
In our case, solving for \( b \) gives us:
  • \( b = \frac{2\pi}{52} \approx \frac{\pi}{26} \)
This calculation allows us to fit the sine wave accurately to the sales data over time.
Horizontal Shift
The horizontal shift in a sine wave determines where the wave begins relative to the origin. For soup sales, the peak occurs not at the start of the cycle but in the 5th week, necessitating a horizontal shift.
Typically, a sine wave reaches its first peak at \( \frac{T}{4} \). For a 52-week cycle, this peak would ideally be at week 13. However, since the maximum occurs during the 5th week, we need to adjust the wave's start point:
  • To calculate the shift, set up the equation: \( \frac{b(5 + c)}{b} = \frac{\pi}{2} \)
  • Plugging in \( b = \frac{\pi}{26} \), we find: \( 5 + c = 13 \)
  • Thus, \( c = 8 \)
This means we shift the wave by 8 units to align correctly with the observed sales peak.
Average Value
The average value of a sine wave describes the wave's midline, which is a critical component for modeling purposes. It indicates the baseline around which the wave oscillates.
In the context of soup sales, the average value is the midpoint between the maximum and minimum sales figures. This gives you insight into the general trend over the cycle:
  • Average Value, \( d = \frac{215 + 100}{2} = 157.5 \)
The average value of 157.5 million suggests that over the course of the year, soup sales hover around this number, fluctuating due to seasonal effects.By integrating the average value into the sine function, the model accurately reflects the expected sales trends.

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