Question

There is a ferris wheel of radius 30 feet. When the compartments are at their lowest, it is 2 feet off the ground. The ferris wheel makes a full revolution in 20 seconds. Using a cosine function, write an equation modelling the height of time?

Answer 1

To solve this type of question we use the concepts of ferris wheel, and cosine equation. We use the time period given to solve the constant values in the cosine equation and use the minimum height and diameter of the wheel to find the vertical shift of the ferris wheel. Therefore using all these values we write the equation modelling of time.

Complete step by step answer:
A cosine equation is generally of the form $y = a\cos \left( {b\left( {x - c} \right)} \right) + b$ , where the parameters represent the following:
$|a|$: the amplitude .When it is negative, it defines a reflection in the x-axis.
$\dfrac{{2\pi }}{b}$ Is the period, in this case the length of time it takes for the ferris wheel to come back to its starting point.
$c$ is the phase shift, or the horizontal displacement.
$d$ is the vertical shift
In this case, we can instantly deduce that the period is $20$ seconds. We will therefore solve for
$\dfrac{{2\pi }}{b} = 20$
$2\pi = 20b$
$\Rightarrow b = \dfrac{\pi }{{10}}$
The amplitude will be given by the formula $Amplitude = \dfrac{{\max - \min }}{2}$ .We know the minimum height is $2feet$. Since the radius is $30feet$ ,the diameter measures $60feet$ ,and so the highest point is $62feet$.
The amplitude is therefore $\dfrac{{62 - 2}}{2} = 30$
The vertical transformation is given by $\min + amp$ , or $\max - amp$ which is $2 + 30 = 32$.
Finally, due to the nature of the cosine function, the cosine function always starts at a maximum (except when parameter a is negative, in which case it starts at a minimum). I assume that when the time starts, the people are just getting on, so the ferris wheel will be at a minimum. Therefore, $a \ne 30$ but instead $a = - 30$
Therefore, the equation is $h = - 30\cos \left( {\dfrac{\pi }{{10}}t} \right) + 32$, where $h$ is the height in feet and $t$ is the time in seconds. We finally note the restrictions to be , $\\{ t|t \geqslant 0,t \in \mathbb{R}\\}$ because it is impossible to have a negative period of time. The h value will always be positive, so we don't have to restrict that.

Note:
To solve such problems we suggest you not to focus on the strange names of the appliances given but you need to focus only on the main part from which the answer can be calculated easily. Here we have given the equation of height varying sinusoidally in which time is a variable.