The sales of APEX Toy Company are seasonal. They peak in earlyDecember at $100,0

Question

The sales of APEX Toy Company are seasonal. They peak in earlyDecember at $100,000 and the lowest point is in early June at$20,000.
A. Write the equation of a funtion f(t) which could representthe dollar sales of this company at any time, t, months afterJanuary. Express your equation using radian measures. (Yourequation should be of the form of either f(t)= A sin (B( t + or -C)) + or - D or f(t)= A cos (B(t + or - C)) + or - D
On a separate sheet of graph paper, graph two periods (2years) of the functions using the 12 months of the year (notnumbers, and use them twice!) as your horizontal axis.
B. What would you expect the sales to be in early APriL?
C.In what months can Apex expect sales to be $60,000?
A. Write the equation of a funtion f(t) which could representthe dollar sales of this company at any time, t, months afterJanuary. Express your equation using radian measures. (Yourequation should be of the form of either f(t)= A sin (B( t + or -C)) + or - D or f(t)= A cos (B(t + or - C)) + or - D
On a separate sheet of graph paper, graph two periods (2years) of the functions using the 12 months of the year (notnumbers, and use them twice!) as your horizontal axis.
B. What would you expect the sales to be in early APriL?
C.In what months can Apex expect sales to be $60,000?

Explanation / Answer

if the high is 100,000 and the low is 20000, what is one-half ofthe difference? this is the amplitude. if we add the amplitude to the low value, we get the value of thevertical shift. or subtract the amplitude from the high value and we should get thesame answer. Since the high point is near the start of a year and the next highis close to the end of a year and the low is near the middle of theyear, high to low to high is more of a cosine curve . if we consider numbering the x axis with 0 akin to Jan 1, then 12 is akin to Jan 1 of the next year. to have horizontal shift so that the peak is at early December, weneed to shift left 0.75.   y = vertical shift + amplitude * cos(B(t+0.75) to find B, we need to have 2pi/B =12, so B = 12/(2pi) put it all together and you get the equation of the curve. to have two cycles, you want to have your graph start at beforedecember and the y axis at Jan and go for another 24 months. the scale of the y-axis must be large, you need to show 100,000units high.

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