## Answers to ExercisesforSection 1: Modeling with the Sine Function

In the following graphs, the first-named function appears in red; the second-named function in yellow.

1.

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7.

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13.

 15. $f(x) = 1 + \sin(2\pix)$ 17. $f(x) = 1.5\sin(4\pi(x-0.25))$ 19. $f(x) = 50\sin(\pi(x-5)/10) - 50$

21. (a) Maximum sales occurred when $t \approx 4.5$ (during the first quarter of 1996). Minimum sales occured when $t \approx 2.2$ (during the third quarter of 1995) and $t \approx 6.8$ (during the third quarter of 1996). (b) Maximum quarterly revenues were $0.561$ billion; minimum quarterly revenues were $0.349$ billion. (c) maximum: $0.455 + 0.106 = 0.561;$  minimum: $0.455 - 0.106 = 0.349$

23. $s(t) = 7.5\sin(2\pi(t-9)/12) + 87.5$

25. $d(t) = 5\sin(2\pi(t-1.625)/13.5)+10$

27. (a) $u(t) = 2.5\sin(2\pi(t-0.75)) + 7.5$    (b) $c(t) = 1.04^t[2.5\sin(2\pi(t-0.75)) + 7.5$

29. Amplitude$= 0.106,$ vertical offset$= 0.455,$ phase shift$= 1.16,$ angular frequency$= 1.39,$ period$= 4.95.$ In 1995 and 1996, quarterly revenue from the sale of computers at Computer City fluctuated in cycles of $4.95$ quarters about a baseline of $0.455$ billion. Every cycle, quarterly revenue peaked at $0.561$ billion ($0.106$ above the baseline) and dipped to a low of $0.349$ billion. During the second quarter of 1995 $(t = 1.16)$ quarterly revenues were was at the baseline level and on an upward cycle.

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 Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: September, 1996