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 $ 42000 in April and a minimum of about $ 25000 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 corresponds to the month.

Answer 1

Answer:

$f(x)=8500\sin\left(\frac{\pi}{6}\left(x-1\right)\right)+33500$

Step-by-step explanation:

Given information:

Maximum revenue = 42000

Minimum revenue = $25000

Time period = 12 months

We need to write a function that models the boutique's revenue during the year, where corresponds to the month.

$f(x)=Asin(B[x-C])+D$           .... (1)

where, A is amplitude, $\frac{2\pi}{B}$ is period, C is phase shift and D is midline.

$A=Amplitude =\frac{Maximum-Minimum}{2}\Rightarrow \frac{42000-25000}{2}=8500$

$D=midline =\frac{Maximum+Minimum}{2}\Rightarrow \frac{42000+25000}{2}=33500$

$Period=\frac{2\pi}{B}$

$12=\frac{2\pi}{B}\Rightarrow B=\frac{\pi}{6}$

Substitute the value of A, B and D in equation (1).

$f(x)=8500\sin\left(\frac{\pi}{6}\left(x-C\right)\right)+33500$         ..... (2)

In April, revenue of the store is $42000. So, the graph passes through the point (4,42000).

$42000=8500\sin\left(\frac{\pi}{6}\left(4-C\right)\right)+33500$

$42000-33500=8500\sin\left(\frac{\pi}{6}\left(4-C\right)\right)$

$8500=8500\sin\left(\frac{\pi}{6}\left(4-C\right)\right)$

Divide both sides by 8500.

$1=\sin\left(\frac{\pi}{6}\left(4-C\right)\right)$

$\sin \frac{\pi}{2}=\sin\left(\frac{\pi}{6}\left(4-C\right)\right)$

On comparing both sides we get

$\frac{\pi}{2}=\frac{\pi}{6}(4-C)$

$3=4-C$

$C=4-3$

$C=1$

Substitute the value of C in equation (2).

Therefore, the required function is $f(x)=8500\sin\left(\frac{\pi}{6}\left(x-1\right)\right)+33500$.

Answer 2

The sinusoidal function that models the revenue, in thousands, is given by:

$f(x) = 8.5\sin{\left(\frac{\pi}{6}\right[x - 1])} + 33.5$

The standard form of the function is:

$f(x) = A\sin{(B[x - C])} + D$

In which:

• The amplitude is $2A$.
• The period is $\frac{2\pi}{B}$.
• The horizontal shift is C.
• The vertical shift is D.

In this problem, first, we consider that the revenue is between $25,000 and $42000, thus, the amplitude is of $17,000, then:

$2A = 17$

$A = \frac{17}{2}$

$A = 8.5$

In a standard equation, this means that the function would assume values between -8.5 and 8.5. We want them to be between 25 and 42, thus, a vertical shift of $D = 25 - (-8.5) = 33.5$ is used.

We want a period of 12 months, which is 1 year, thus:

$\frac{2\pi}{B} = 12$

$12B = 2\pi$

$B = \frac{2\pi}{12}$

$B = \frac{\pi}{6}$

Using a graphing calculator, the maximum revenue is for $x = 3$ and the minumum revenue is for $x = 9$. We want it for $x = 4$ and $x = 10$, so a shift to the right of 1 unit, thus the horizontal shift is $C = 1$

Then, the equation is:

$f(x) = 8.5\sin{\left(\frac{\pi}{6}\right[x - 1])} + 33.5$

Which graph is sketched at the end of this answer.

A similar problem is given at https://brainly.com/question/22136310