a Use a unit circle to find the height to the center. Remember that the unit circle has a radius of 1 while the radius of our Ferris Wheel is 30.
B
b The maximum distance is either directly above or directly below the center.
C
c Consider the function y=sin x.
A
a See solution.
B
b 30'
C
c y=30sin x
Practice makes perfect
a Notice that we are looking for the vertical distance between the seat and the center, where the center represents y=0 in a coordinate plane.
Therefore, we should identify the y-coordinate of each position around the Ferris wheel. To do this we can use the unit circle. Notice that - 45^(∘) and 315^(∘) have the same relative positions on the unit circle. This is also true for a rotation of 0^(∘) and 360^(∘).
As previously argued, the unit circle has a radius of 1 unit. Therefore, to determine the position from the center of the Ferris wheel, we have to multiply each y-coordinate with its radius.
r|rcl
- 45^(∘) & - 1/sqrt(2)(30) -5pt & = & -5pt - 21.21 feet [1em]
0^(∘) & 0(30) -5pt & = & -5pt 0 feet [0.5em]
45^(∘) & 1/sqrt(2)(30) -5pt & = & -5pt 21.21 feet [1em]
90^(∘) & 1(30) -5pt & = & -5pt 30 feet [0.5em]
135^(∘) & 1/sqrt(2)(30) -5pt & = & -5pt 21.21 feet & [1em]
180^(∘) & 0(30) -5pt & = & -5pt 0 feet [0.5em]
225^(∘) & - 1/sqrt(2)(30) -5pt & = & -5pt - 21.21 feet [1em]
270^(∘) & - 1(30) -5pt & = & -5pt - 30 feet [0.5em]
Now we can complete the table.
|l|c|c|c|c|c|c|c|c|c|
x (angle) & - 90^(∘) & - 45^(∘) & 0^(∘) & 45^(∘) & 90^(∘) & 135^(∘) & 180^(∘) & 225^(∘) & 270^(∘)
y (height) & - 30' & - 21.21' & 0' & 21.21' & 30' & 21.21' & 0 & - 21.21' & - 30'
b When the Ferris Wheel is directly above or below the center, the seat is at its maximum vertical distance from the center. In these positions the wheel's radius becomes the vertical distance between the center and the seat. Therefore, the maximum distance is 30 feet.
c Let's plot the points from Part A in a coordinate plane.
Finally, we will connect the points with a smooth curve.
Notice that this curve resembles a sine curve, but with an amplitude of 30. Therefore, the equation that fits the County Ferris Wheel ride is y=30sin x.
