To make sure the flip is in unison, after 4 seconds should the flip occur, as given by the option C: 4 seconds.
What does synchronized motion mean?
Synchronized motion is a motion of two or more system which goes in syncronization with each other.
For this case, the flip will occur such that both gymnasts are synchronized and flip is unison (simultaneous).
For that, we need the time(x) and the height (y) same.
Since the motion paths are:
[tex]y= -(x-5)^2 + 3\\\\y = -3(x-3)^2 + x + 1[/tex]
Therefore, we need to find the values of x for which same y comes as output.
Let x = t be the time for which the height y is same, then:
[tex]-(t-5)^2 + 3 = -3(t-3)^2 + t + 1[/tex] = height achieved by both gymnasts.
Solving it further:
[tex]-(t-5)^2 + 3 = -3(t-3)^2 + t + 1\\3(t^2 + 9 -6t) -(t^2 + 25 -10t) -t + 2 = 0\\\\2t^2 -9t + 4 = 0\\2t^2 -8t - t + 4 = 0\\2t(t-4) -1(t-4) = 0\\(2t -1)(t-4) = 0\\\\t = 4, t = 1/2[/tex]
Assuming they planned to do flip after integral number of time in seconds, the correct option is t = 4 seconds.
Thus, to make sure the flip is in unison, after 4 seconds should the flip occur, as given by the option C: 4 seconds.
Learn more about solving system of equations here:
https://brainly.com/question/2825832
