finding coefficients in a trigonometric function (ferris wheel question)

jbudd

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Jan 16, 2018
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Hi,

This is a ferris wheel question, which I believe is quite common in trig lessons. The wheel is 135 meters high, and turns at a constant speed of 30 minutes for one complete cycle.
The height as a function of time can be written as h(t) = a + b cos(ct). I need to find the coefficients for a, b and c. C is in radians and the wheel starts moving at t=0.

I can get as far as solving for c, which works out to be pi/15. But I am struggling with a and b.

I have been told that h(0) = 0: a + b = 0. And that h(15) = a - b = 135. But I don't understand why these statements are true. For example why do we add a + b in the first statement when h(0) = 0, and why do we subtract b from a in the second statement?

I'm also told that a = - b = 135/2. But I 'm not sure why we have to divide 135 by 2.

Any explanation would be appreciated. As you can see I've been given the answer, but I don't fully understand the steps to arrive at the answer.

Thank you.
 
You haven't actually stated the problem. Did you read the Read before posting announcement? "Post the complete text of the exercise. This would include the full statement of the exercise and its instructions, so the tutors will know what you are working on."

I can guess that you mean this:

A ferris wheel is 135 meters high, and turns at a constant speed of 30 minutes for one complete cycle. The height of a seat as a function of time can be written as h(t) = a + b cos(ct), where t is the time in minutes. Find the coefficients a, b and c. Parameter c is in radians per minute and the wheel starts moving at t=0 (that is, at that time the seat is at ground level, h = 0).

You found that c = pi/15, so that when t = 30, ct = 2 pi and the wheel has made one full turn. That is correct.

Why divide 135 by 2? Because 135 is the diameter (in meters), so half of that is the radius. The cosine function varies between 1 and -1, so you multiply it by the radius. But this is a different way to solve the problem than what you were shown.

You were given two equations: h(0) = 0: a + b = 0, and h(15) = a - b = 135. The first is because when t = 0, cos(ct) = cos(0) = 1, so h(0) = a + b cos(0) = a + b. The second is because when t = 15, ct = pi/15 * 15 = pi, and cos(pi) = -1, so h(15) = a + b cos(-1) = a - b.

Can you work out the rest? That's a matter of solving the system of linear equations, which is easily done by addition.
 
thank you

Thank you. That does help.

Regarding the rules of posting, I will be more careful to post the complete text of the problem next time. In my haste, I left out a couple points that I thought weren't critical, since it was mostly just a few parts of the wider problem that I was struggling with. Anyway, point well taken.

Thanks again.
 
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