multi step calculus

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 205 million during the 5th week of the year and at a minimum of 100 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		million cans

(c) Use the constants from parts (a) and (b) to construct the sine model for weekly soup sales in million cans on the xth week of the year, with data from 0 ≤ x ≤ 52.

(Round all numerical values to three decimal places.)

f(x) =

Answer 1

The period is twice as long as the max to the min. The max is at 5 week, the min at 31 weeks so the time between them is 26 weeks and two times that is 52 weeks. Since the period is 52 weeks, the value of "b" is 2π/52 = π/26 which is approximately 0.121

A sine function starts at zero and goes up. This function starts to the right of zero but reaches a max at 5 weeks. The time between the max and zero is 1/4 of the period (so 13 weeks). So the place where this function hits zero is 13 weeks before the "week 5" max which is at - 8 weeks. So the horizontal shift is -8 weeks. To get the proper value of "c" we would need to list it as +8 because the horizontal shift requires taking the opposite value but, the way the generic formula is written, we also have to multiply it by the "b" value of π/26 making c = 8π/26 = 4π/13 which is approximately 0.967

The vertical shift (which is the average value) is halfway between the max and min so (205 + 100)/2 = 152.5 million cans of soup. The amplitude is then the max - average (vertical shift) value = 205 - 152.5 = 52.5 million cans of soup.

The model is then f(x) = 52.5sin(0.121x + 0.967) + 152.5