Daniel B. answered • 12/16/21
Experienced, patient tutor for mathematics and other subjects
To answer the first question, we must know the period of this function -- that is, how often it repeats. That's how long the ferris wheel takes to make one circle. We also need to know whether we're using degrees or radians here. I'm going to guess degrees -- otherwise the ferris wheel would be scary fast :-D
Since there are 360 degrees in a circle, we have cos(x) = cos(x + 360), so the period of cos(x) would be 360. If we have cos(2x), then it's going twice as fast, so it takes half as long -- the period of cos(2x) would be 1/2 * 360 = 180. And the coefficient of x (or t or whatever) inside the cosine is the *only* number that matters for period --- the period of 17cos(2x) - 3 would still be 180, for example.
In this problem, our cosine looks like cos(2.4t), so it's going 2.4 times as fast as cos(t) -- the period is (1/2.4)*360 = 150 seconds per revolution. For five turns, the wheel takes 5 * 150 = 750 seconds.
I'm not sure what you mean by "equation of the axis" -- should that be "axes of the equation"? We have the t axis, for time in seconds; and the h axis, for height above ground.
What is Ana's maximum height? Well, our equation is h(t) = -4 * cos(2.4t) + 6. We know that the cosine goes from -1 to 1, so the height goes from ( - 4)*1 + 6 = 2m at bottom, to a top height of ( - 4)*( - 1) + 6 = 10m.
To discover when she reaches that maximum height, we can set h = 10 and solve for t --- but we already know that the max height happens when cos(2.4t) = -1, so in fact it's even easier:
cos(2.4t) = -1
⇒ 2.4t = 180 (the *first* time she's at max height)
⇒ t = 75 seconds.
So the first time is at 75 seconds, and then she's there again every time the wheel turns--that is, every 150 seconds.
Cheers!
