a ferris wheel has the diameter of 240 feet and the bottom of the ferris wheel is 9 feet above the ground. find the equation of the wheel if the origin is placed on the ground directly below the center of the wheel
If a circle of radius r has center at (h,k) the equation of the circle is
(x-h)² + (y-k)² = r²
So, plug and chug:
x² + (y-(120+9))² = 120²
x² + (y-129)² = 14400
To find the equation of the ferris wheel, let's first set up a coordinate system with the origin at the ground directly below the center of the wheel.
Since the diameter of the ferris wheel is 240 feet, the radius is half of that, which is 240 / 2 = 120 feet.
With the bottom of the ferris wheel 9 feet above the ground, the center of the wheel is located at (0, 9 + 120) = (0, 129).
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values we have, the equation of the ferris wheel is:
(x - 0)^2 + (y - 129)^2 = 120^2
Simplifying further, the equation of the ferris wheel is:
x^2 + (y - 129)^2 = 14400
To find the equation of the Ferris wheel, we need to consider the geometry of a circle. The equation of a circle with its center at the origin and a radius of 'r' can be given by the equation:
x^2 + y^2 = r^2
In this case, since the Ferris wheel has a diameter of 240 feet, the radius would be half of that, which is 240/2 = 120 feet.
However, since the bottom of the Ferris wheel is 9 feet above the ground, we need to offset the y-coordinate accordingly. We'll shift the entire circle vertically up by 9 feet.
So the equation of the Ferris wheel can be written as:
x^2 + (y-9)^2 = 120^2
This equation represents the Ferris wheel in the xy-coordinate system, with the origin placed on the ground directly below the center of the wheel.