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.

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

The standard form of the function is:

In which:

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

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 is used.

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

Using a graphing calculator, the maximum revenue is for and the minumum revenue is for . We want it for and , so a shift to the right of 1 unit, thus the horizontal shift is

Then, the equation is:

Which graph is sketched at the end of this answer.

A similar problem is given at link

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.

          .... (1)

where, A is amplitude, is period, C is phase shift and D is midline.

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

        ..... (2)

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

Divide both sides by 8500.

On comparing both sides we get

Substitute the value of C in equation (2).

Therefore, the required function is .

3

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240