# Urgent! 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?

## Help! Radius - 30 feet Lowest point - 2 feet Time for 1 revolution - 20 seconds Write cosine function!

Dec 3, 2016

$h = - 30 \cos \left(\frac{\pi}{10} t\right) + 32$, {t|t ≥ 0, t in RR}

#### Explanation:

An equation in cosine is generally of the form $y = a \cos \left(b \left(x - c\right)\right) + d$, where the parameters represent the following:

$| a |$: the amplitude. When it is negative, it denotes a reflection in the x axis.

$\frac{2 \pi}{b}$ 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 $b$.

$\frac{2 \pi}{b} = 20$

$2 \pi = 20 b$

$b = \frac{2 \pi}{20}$

$b = \frac{\pi}{10}$

The amplitude will be given by the formula $\frac{\text{max" - "min}}{2}$. We know the minimum height is 2 feet. Since the radius is 30 feet, the diameter measures $60$ feet, and so the highest point is at $62$ feet.

The amplitude is therefore $\frac{62 - 2}{2} = \frac{60}{2} = 30$.

The vertical transformation is given by $\min + a m p$, or $\max - a m p$, 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 $a \ne - 30$.

Therefore, the equation is $h = - 30 \cos \left(\frac{\pi}{10} t\right) + 32$, where $h$ is the height in feet and $t$ is the time in seconds. We finally note the restrictions to be {t|t ≥ 0, t in RR}, 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.

Hopefully this helps!