Question

A Ferris wheel has a diameter of 191 feet and sits 12 feet above the ground. It rotates in a counterclockwise direction, making one complete revolution every 25 minutes. Place your coordinate system so that the origin of the coordinate system is on the ground below the bottom of the wheel. Find the equation of the Ferris wheel.

Answer 1

The equation of the Ferris wheel is
`y=-95.5\cos ((2\pi)/(25)x)+107.5`.

Explanation:

The general form of cosine function is

`y=A\cos (Bx+C)+D` .... (1)

where, A is amplitude, period is
`(2\pi)/(B)`, C is phase shift and D is midline.

It is given that a Ferris wheel has a diameter of 191 feet and sits 12 feet above the ground. So, the minimum value of the function is 12 and the maximum value of the function is 191+12=203.

The midline of the function is

`D=Midline=(Maximum+Minimum)/(2)=(203+12)/(2)=107.5`

It makes one complete revolution in every 25 minutes.

`Period =25`

`(2\pi)/(B)=25`

On cross multiplication we get

`(2\pi)/(25)=B`

Phase shift is not given, So C=0.

Substitute
`B=(2\pi)/(25)`, C=0 and D=107.5 in equation (1).

`y=A\cos ((2\pi)/(25)x+0)+107.5`

`y=A\cos ((2\pi)/(25)x)+107.5` .... (2)

It is given that the origin of the coordinate system is on the ground below the bottom of the wheel. It means the graph passes through the point (0,12).

Put x=0 and y=12 in equation (2) to find the value of A.

`12=A\cos ((2\pi)/(25)(0))+107.5`

`12=A(1)+107.5`

`12-107.5=A`

`-95.5=A`

Substitute A=-95.5 in equation (2).

`y=-95.5\cos ((2\pi)/(25)x)+107.5`

Therefore the equation of the Ferris wheel is
`y=-95.5\cos ((2\pi)/(25)x)+107.5`.