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Question: Suppose you wanted to model a Ferris wheel using a sine function that took $60$ seconds to complete one revolution. The Ferris wheel must start $0.5\,\textrm{m}$ above ground. Provide an equation of such a sine function that will ensure that the Ferris wheel's minimum height of the ground is $0.5\,\textrm{m}$. Explain why your equation works.

Format of equation: $y = a\sin k(x - d) + c$

I need help because I don't understand how to determine the equation.

For example: how am I suppose to find the amplitude and the $c$ value if I only have the minimum value and not the maximum value? Also, I don't understand how to find the phase shift (value of $d$) if I am only given the minimum value and the period. Please help!

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    $\begingroup$ Indeed we do not know the radius (or diameter) of the wheel. So let's say that the radius is $R$. The wording of the problem is not very clear; I will assume that the function is to represent the vertical component of a point along the edge on the wheel, in meters. With this, we want a function that has a minimum value of $0.5$ and a maximum value of $0.5+2R$. The time frame probably doesn't matter. Therefore, we want a sine function of the form $$ y = 2R \frac{\sin (kt)+1}{2} + 0.5 $$ Why does this work? Note that $\frac{\sin (kt)+1}{2} $ is always between $0$ and $1$... $\endgroup$ – Matti P. Jul 16 '20 at 7:21
  • $\begingroup$ Can you continue from that? $\endgroup$ – Matti P. Jul 16 '20 at 7:22
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Since the question states that the Ferris Wheel must start half a meter off of the ground, then we can make our phase shift $d=0$. This allows us to assume that the minimum height is achieved at $x\equiv \frac{\pi}{2}n$ where $n$ is every other odd integer starting with $n=3$. This is because the sine function is $-1$ at those values, and is at a minimum.

Next, sine functions ,$y= a\sin{k(x-d)}+c$, are $2\pi$ periodic, meaning that it takes $2\pi$ radians, or $1$ period, to get back to your initial starting point. The period, $T$, is given as $60$ seconds. Using the formula for the period of a sine and cosine function, $T=\frac{2\pi}{|k|}$, we find that $|k|=\frac{\pi}{30}$. The absolute value signs are not really necessary, but period is typically always positive and $k$ can be positive or negative.

Now to find the amplitude. No speed was specified, nor was the radius of the Ferris Wheel, and the only way I see to solve this is to let $a=r$ where $r$ is the radius of the Ferris Wheel.

Finally, we need that when $\sin{k(x-d)}=-1$, $y=0.5$. Setting $y=-r+c=0.5$, we see that $c=r+0.5$.

$$y=r\sin{\frac{\pi}{30}x} \,+ r+\frac{1}{2}$$

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