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 $ 59000 in April and a minimum of about $ 29000 in October. 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. If needed, you can enter π=3.1416... as 'pi' in your answer.

Answer:[tex]f(x) = -15000\sin [\frac{\pi}{6}(x-7)] + 44000[/tex]Step-by-step explanation:Given, the function that shows the revenue.[tex]f(x) = A\sin [B(x-C)] + D[/tex],Where, x corresponds to the month.∵  Revenue reaches a maximum of about $ 59000 in April and a minimum of about $ 29000 in October,i.e. the maximum value f(x) is $ 59000 when x = 4,And,  the minimum value f(x) is $ 29000 when x = 10,So, the amplitude, [tex]A = \frac{max - min}{2} = \frac{59000 - 29000}{2} = \frac{30000}{2}=15000[/tex][tex]D = \frac{max + min}{2}=\frac{59000 + 29000}{2} = \frac{88000}{2}=44000[/tex]The minimum of sine function corresponds to [tex]-\frac{\pi}{2}[/tex], here it is 10 and maximum [tex]\frac{\pi}{2}[/tex], here it is 4.Period = 12 months,But we know period = [tex]\frac{2\pi}{B}[/tex][tex]\implies \frac{2\pi}{B} = 12[/tex][tex]\implies B = \frac{\pi}{6}[/tex]∵ [tex]\frac{4+10}{2} = \frac{14}{2}= 7[/tex],Thus, f(x) is symmetrical about x=7,⇒ C = 7,Also, f(x) is minimum at x = 10,So, A = - 15000,Hence, the required function would be,[tex]f(x) = -15000\sin [\frac{\pi}{6}(x-7)] + 44000[/tex]