Trigonometry problem

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.

Answer 1

Elaina

So ... the minimum height is 2 cm, the maximum is 200 cm, the first maximum is at t=6 s, and the period of the function is 10 s (because he reaches a maximum every 10 s).

The general equation for a sine function is a sin b(x+c) + d.

Amplitude is a = half distance from minimum to maximum, so here (200 + 2) / 2 = 101

period is 2π / b. As your period here is 10, that means b = 2π/10

c and d are phase shifts. Consider the sine function just using a and b so 101 sin (2π/10) x.

This will have the correct amplitude and period but we need to shift it so that the first maximum of the function after t=0 passes through (6, 200), which is your first maximum from the data.

The maximum of the sine function just given will be y=101, when (2π/10) x = π/2 ie x = 10/4 = 2.5

So the maximum of this one will be at (2.5, 101) and we need it at (6,200).

So we need to shift the sine function 6 - 2.5 = 3.5 units right, and 200 - 101 = 99 up.

So c will be -3.5 (sign has to be negative for a right shift) and d will be + 99

So your function should be 101 sin ((2π/10)(x-3.5)) + 99.

Suggest you plot it on Desmos or a graphing calculator and check it is correct !

Mike