Respuesta :
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]
