# Trigonometry Ferris Wheel Question

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!

• 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 $1$... – Matti P. Jul 16 '20 at 7:21
• Can you continue from that? – Matti P. Jul 16 '20 at 7:22

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 $$c=r+0.5$$.
$$y=r\sin{\frac{\pi}{30}x} \,+ r+\frac{1}{2}$$