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

Hello again!
This is similar to the question we looked at yesterday, which also related to sinusoid wave functions.
The principles are the same; we must find the amplitude, frequency, and (additionally this time) the displacement of the wave compared to the original sin wave.
As a reminder, the sin function will take the form:
f(x) = a*sin(b*x + c) + d
Where "a" represents the amplitude, "b" represents the frequency, and "c" and "d" represent the horizontal and vertical displacement, respectively.

To get started, the frequency is more or less given to us; "Owen reaches a maximum height... every 10 seconds", which means that a full wave completes in 10 seconds, compared to a full wave every 2*pi units in the base sin function, so consider the multiplication you'd have to do to convert 2*pi to 10. (This should give us a hint about the value of "b" in the equation)
We can also find the amplitude by taking the difference between the maximum and minimum heights; the maximum height is 200 cm, and the minimum is -2 cm, for a difference of 202 cm. The amplitude would be half this amount, 101 cm. (This should give us a hint about the value of "a" in the equation)
Lastly, for displacement. Normally, a sin function without any displacement would start at the origin (0,0) and reach its peak in half a period (5 seconds, since this wave has a period of 10 seconds). The question says "Owen reaches his first maximum in 6 seconds", which means that the wave should be displaced to the left by 1 second (a hint for the value of "c")
Also, the maximum and minimum heights are not symmetrical around 0, like a default sin function. Rather than a vertical range of -101 to 101, this wave has a range of -2 to 200, which seems like a vertical displacement upward of 99 units (a hint for the value of "d").

"Using the template for the sin function and the information about the wave given, please try to fill in the values of a, b, c, and d to get the appropriate function to represent the wave," is another way you can think about this question.

If you have any additional questions or concerns, please feel free to reach out, and I will try my best to address them as soon as I can :)

Answer 2

Amendment: I was wrong about the hint for value "c"; the wave should be shifted to the *right* by 1, not to the left, in order to be consistent with the description in the question. My apologies.