A ferris wheel with a diameter of 100 feet rotates at a constant rate of 4 revolutions per minute. Let the center of the ferris wheel be at the origin.

1. Each of the ferris wheel's cars travels around a cirlce.
a) Write an equation of the circle where x and y are measured in feet.
b)sketch a graph of an equation you wrote in part (a)
c) Use the vertical line test to determine whether y is a function of x.
d) What does your answer to part (C) mean in the context of the problem?

1a. (x-h)^2 + (y-k)^2 = r^2.

C(h,k) = C(0,0).

Eq: x^2 + Y^2 = (50)^2 = 2500.

b. Use these points for graphing:
Angle (X,Y)
0. (5,0)
45. (3.54,3.54)
90. (0,5)
135 (-3.54,3.54)
180 (-5,0)
225 (-3.54,-3.54)
270 (0,-5)
315 (3.54,-3.54)

Sample Calculations
Scale for radius(r) = 0.1 * 50 = 5Ft.

X = r*cosA = 5*cos(0) = 5.
Y = 5*sin(0) = 0.

X = 5*cos45 = 3.54.
Y = 5*sin45 = 3.54.

a) The equation of the circle can be written as:

x^2 + y^2 = (50)^2

b) Here is a sketch of the graph:

(0,50) (50,0) (-50,0)
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(0,-50) (0,0) (0,0)

c) The vertical line test states that if a vertical line intersects a graph in more than one point, then the graph does not represent a function. In this case, if we draw a vertical line at any x value (except 0), it will intersect the graph in two points. Hence, y is not a function of x.

d) This means that at each x-coordinate (except at the origin), there are two possible y-values. In the context of the problem, it implies that multiple cars of the ferris wheel can be found at the same x-coordinate, occupying different y-values simultaneously.

a) The equation of the circle with the center at the origin and a diameter of 100 feet can be written as: x^2 + y^2 = (100/2)^2

b) Here is a sketch of the graph of the equation:

[Graph of circle with center at the origin and radius of 50 feet]

c) The vertical line test is a way to determine if each x-coordinate has a unique y-coordinate. In this case, since the circle equation represents a perfectly round ferris wheel, for each x-coordinate, there will be two possible y-coordinates. Therefore, y is not a function of x.

d) In the context of the problem, this means that each x-coordinate on the ferris wheel, which represents the horizontal position of a car, has two possible y-coordinates (vertical positions). Hence, multiple cars can occupy the same horizontal position on the ferris wheel at the same time.

To answer the question, let's break it down step by step:

a) To write an equation of the circle, we need to determine its center and radius. Since the center of the ferris wheel is at the origin, the center coordinates would be (0, 0). The radius of the circle can be calculated as half the diameter, which in this case would be 100/2 = 50 feet.

Using the formula for the equation of a circle, (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius, we can substitute the values to get the equation:

(x - 0)^2 + (y - 0)^2 = 50^2
x^2 + y^2 = 2500

Therefore, the equation of the circle where x and y are measured in feet is x^2 + y^2 = 2500.

b) Sketching the graph of this equation, we can draw a circle centered at the origin with a radius of 50 feet. The graph would show all the possible locations of the cars as they travel around the circle.

c) To determine whether y is a function of x, we can use the vertical line test. The vertical line test states that if any vertical line intersects the graph of a relation (in this case, the circle), at more than one point, then the relation is not a function.

In the graph of the equation, we can see that for each x-coordinate, there are two possible y-coordinates (positive and negative) on the circle. This indicates that y is not a function of x.

d) In the context of the problem, the result of part (c) means that for any given x-coordinate on the circle, there are two possible heights (y-coordinates) that the ferris wheel car can be at. This is because the ferris wheel rotates at a constant rate, and at a particular x-coordinate, there are two cars on opposite sides of the circle. Therefore, y represents the height of the car, but it is not uniquely determined by x alone.