Sales of ski equipment varies through the year from highs in January & December of $20 million, to a low in June of $2 million. Write a trigonometric function that describes this cycle. Calculate the expected sales for April.

I think I got the first part:
9cos(pi/6) + 11

I am just not sure how to calculate expected sales for April?

Your "function" contains no variables!

don't you mean

y = 9cos(pi/6)t + 11 ??

so if June corresponds with t=6 and Dec corresponds with t = 12
then April would match with t = 4
and y = 9cos(4pi/6) + 11
= 6.5

BTW, January does not seem to fit into the cycle. How can the max be in Dec and in Jan ?

A function generally implies the inclusion of one or more independent variables, which in this case, is t that represents the month.

So you'd need a function that has a cycle of 12 months, a maximum of 20 and a minimum of 2.

The choice of cosine is excellent.

The multiplicative constant of 9 and the additive constant of 11 is also correct.

So if you write the function as
S(t) = 9cos(pi*t/6) + 11
where t=0 to 12. (0 for the beginning of January, 12 for the end of december).
So S(0)=S(12)=20, and S(6)=2.

Try to figure out the sales for April (fourth month of the year).

I think you mean

9 cos (pi * x /6)

where x is the month number 0 through 11. And very neat that snswer is. It provides a low of 2 and a high of 20.

Mind you, you might want to mention that your actual answer is:

1,000,000(9cos(pi/6) + 11)

or

10^6(9cos(pi * x/6) + 11)

to make up that factor of a million.

So what about April? That's month 3 in the 0-11 sequence, so its value would be

10^6(9cos(pi * x/6) + 11)

or 11 million.

Oh yeah, I forgot to put the 'x' into my equation. I just plugged in the 4 where the x should be and got 6.5, or as it applies here, 6.5 million. Hope that's right. Thank you all! :)

yeah part exams library go there Pre-Calculus exam

To calculate the expected sales for April, we need to know the phase shift of the trigonometric function. The given information states that the highest sales occur in January and December, which correspond to the peaks of the function. We can assume that January corresponds to a phase shift of 0, and thus December corresponds to a phase shift of 11.

Since the lowest sales occur in June, which is 6 months away from December, we can infer that June corresponds to a phase shift of pi or 6 months. Therefore, April, which is 2 months away from June, will have a phase shift of pi/6 or 2/6.

Now let's write the trigonometric function that describes this cycle considering the phase shift:

S(t) = A * cos(Bt - C) + D

Where:
- A represents the amplitude (the difference between the maximum and minimum values), which is (20 - 2) / 2 = 9.
- B represents the frequency (how many cycles occur in a given interval). Since the cycle repeats every 12 months, B = 2pi / 12 = pi / 6.
- C represents the phase shift we calculated earlier, which is pi / 6. (This is the angle inside the cosine function.)
- D represents the vertical shift (the average value), which is (20 + 2) / 2 = 11.

Now let's calculate the expected sales for April:

S(April) = 9cos[(pi / 6) * (2/6)] + 11
S(April) = 9cos(pi / 18) + 11

The answer is:
S(April) = 9cos(pi / 18) + 11 ≈ 13.02 million dollars.

Therefore, we can expect the sales for April to be approximately 13.02 million dollars.