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!

  • 1
    $\begingroup$ 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

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 $c=r+0.5$.

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.