Question

Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about $54,000 in May and a minimum of about $20,000 in November. Suppose the months are numbered 1 through 12, and write a function of the form f(x)=Asin(B[x−C])+D that models the boutique's revenue during the year, where x corresponds to the month.

A) f(x)=17,000sin( π/6 [x−5])+37,000
B) f(x)=17,000sin( π/6 [x−4])+37,000
C) f(x)=17,000sin( π/6 [x−3])+37,000
D) f(x)=17,000sin( π/6 [x−2])+37,000

Answer 1

The function that models the boutique's revenue during the year is f(x) = 17,000sin(pi/6[x-4]) + 37,000.

Step-by-step explanation:

The function that models the boutique's revenue during the year is:

f(x) = 17,000sin(π/6[x-4]) + 37,000

To determine this, we observe that the revenue reaches a maximum of $54,000 in May and a minimum of $20,000 in November. The maximum value occurs in month 5, so the value of C is 5. The maximum revenue value is $54,000, so the value of D is 54,000. The amplitude of the function, A, can be determined by finding the difference between the maximum and minimum values, which is (54,000 - 20,000) = 34,000. Finally, the period, determined by B, can be found by dividing the number of months in a year (12) by the number of cycles between the maximum and minimum values (1). Therefore, B = 12/1 = 12. Substituting these values into the given equation gives us the final function that models the boutique's revenue.