Question

Canned soups are consumed more during colder months and less during warmer months. A soup company estimates its sales of 16-oz cans of condensed soup to be at a maximum of 215 million during the 5th week of the year and at a minimum of 105 million during the 31st week of the year. (Note: This exercise corresponds to the subsection Characteristics of a Sine Function and requires use of the formulas for average value, amplitude, period, and horizontal shift to construct sine models rather than curve-fitting using technology.)
(a) Calculate the period and horizontal shift of the soup sales cycle. Use these values to calculate the parameters b and c for a model of the form
f(x) = a sin(bx + c) + d.
(Assume that the horizontal shift is negative and the absolute value is less than the period. Round your answers to three decimal places when appropriate.)
period


weeks
horizontal shift


weeks
b =


c =



(b) Calculate the amplitude and average value of the soup sales cycle.
amplitude



million cans
average value

Answer 1

Answer:

(a) b = π/26

c = 4·π/13

(b) The amplitude = 55 million cans

he average value of the soup sales cycle = 160 million cans

Step-by-step explanation:

(a) The given information are;

The volume of the cans of condensed soup sold = 16-oz

The maximum number of cans sold = 215 million

The time at which the maximum number of cans are sold = The 5th week of the year

The minimum number of cans sold = 105 million

The time at which the minimum number of cans are sold = The 31st week of the year

The function representing the number of cans sold = Sine function

f(x) = a·sin(b·x + c) + d

The period, p = 2π/b

When bx + c = 0

x = -c/b

When bx + c = π/2

x = (π/2 - c)/b = 5

(π/2 - c)/5 = b

When bx + c = 3·π/2

x = (3·π/2 - c)/b = 31

(3·π/2 - c)·5/(π/2 - c) = 31

Solving, we get

c = 4·π/13

b = π/26

(b) The amplitude and average value of soup sales

The amplitude = a =  The distance from the midpoint to the highest or lowest point of the function

∴ a = (215 - 105)/2 = 55 million cans

The average value of the soup sales cycle = d = The midline

d =  105 million + 55 million = 160 million cans.