Trigonometric Functions Question - Sinusoidal Function

Question

Owen is jumping on a trampoline. When his feet hit the deck of the trampoline, the material depresses to a minimum height of 2cm. On average, Owen is reaching a maximum height of 200cm every 10 seconds. Determine the equation of a sinusoidal function that would model this situation, assuming Owen reaches his first maximum at 6 seconds.

How could I answer this question? Thanks.

Answer 1

Y=a sin (b (t - h)) + k which a is half of the domain, b is phase shift and k is vertical shift.

a =(Max-Min)/2 = (200-2)/2 = 99

To find b we have the formula: Period = 2pi/b so

10 = 2pi/b therefore: 10b = 2 pi and b = pi / 5

k = 99 + 2 = 101

Since it goes down first so a is negative. The graph has 1.5 cm shift to left since, if it was supposed to have no horizontal shift then the first max would appear after 10 x 3/4 = 7.5 but the problem says that it reaches the first max after 6 seconds which makes a 1.5 seconds horizontal shift to the left

So y = -99 sin (pi/5 (t + 1.5)) + 101