Ferris wheel Trig Question

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 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 m. Explain your answer.

Hey, I don't really know how to do this question, I mean if we don't have max how do we find any equation. I would appreciate it if you could help me to get the answer. thanks

Answer 1

Let’s start with the standard sine function: $$f(t)=\sin t$$ If the radius of the wheel is $r$, then to adjust the amplitude, i.e. the farthest the wheel can go from its middle position, you need to multiply with $r$: $$f(t)=r\sin t $$ Now, you need to adjust the time period. Note, the period for $\sin (nx)$ is $\frac{2\pi}{n}$. Set $\frac{2\pi}{n} =60 \implies n=\frac{\pi}{30}$. $$f(t)=r\sin\left(\frac{\pi}{30}t\right) $$ To ensure that it attains its minimum value at $t=0$, shift the graph to the right by $\frac{\pi}{2}$: $$f(t)=r\sin\left(\frac{\pi}{30} t -\frac{\pi}{2} \right) $$

Now, you need to add a constant that will take care of the minimum height constraint. It is needed that $f(0) = 0.5$, so that constant is $r+0.5$. This gives the final equation: $$f(t) = r+0.5+r\sin\left(\frac{\pi}{30}t -\frac{\pi}{2} \right) $$ Note: Since $r$ is not given in the question, you might assume $r=1$.

Answer 2

Hint:

You want your height $h$ such that:

$h(t) = a + r\sin(bt + c)$

To find these values:

$r$ should be the radius of the ferris wheel. This would make sense, as the height of someone at the top of the ferris wheel is r + height of center, and r - height of center for being at the bottom

$c$ should be chosen such that when $t=0$, $\sin(bt + c) = -1$. This is because you would be at the bottom of the ferris wheel, so this would be the minimum height you can get with the equation for $h$ above.

$b$ should be picked so the period of the ferris wheel is $60 s$.

Finally, $a$ should be picked so that $h(0) = $ min height

These together should be enough to solve this question