Question

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!

Answer 1

`h = -30cos(pi/10t) + 32`, `{t|t ≥ 0, t in RR}`

Explanation:

An equation in cosine is generally of the form `y= acos(b(x - c)) + d`, where the parameters represent the following:

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

•`(2pi)/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`.

`(2pi)/b = 20`

`2pi = 20b`

`b = (2pi)/20`

`b = pi/10`

The amplitude will be given by the formula `("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 `(62 - 2)/2 = 60/2 = 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!=30` but instead `a!=-30`.

Therefore, the equation is `h = -30cos(pi/10t) + 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!