Sinusoidal Functions

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.

I need help with making the equation.

Answer 1

We can use be either a sine or a cosine function for this. I'll pick cosine since the problem talks about the top of the peak at a certain time and the function that starts at the peak and goes down from there is cosine.

The generic equation for a cosine function is h(t) = A*cos[B(t - C)] + D where "A" is the amplitude, "B" defines the period (where period = 2π/B), "C" is the phase shift or horizontal shift, and "D" is the vertical shift.

Step 1) Determine where the center line is that the sinusoidal oscillates about. This is the "D" value. To do that, just average the highest and lowest points (-2 and 200) so (200+ -2)/2 = 99. So D = 99

Step 2) Determine the amplitude. This is just the difference between the maximum and the center line and it is the "A". The max is 200 and the center line is 99 so A = 101.

Step 3) Determine the "B" value by first determining the period and then using P = 2π/B to calculate "B". They tell us the period is 10 sec so 10 = 2π/B or B = 2π/10 and, reducing, B = π/5.

Step 4) Determine the phase shift (horizontal shift), They tell us what this is. C = 6 sec.

Putting these all together we get: h(t) = 101*cos[π/5(t - 6)] + 99 where t is time in seconds and h(t) is height in cm.

You can double check this by plugging this function into your graphing calculator (making sure you are in radians) and it should look like this: