A ferris wheel has a diameter of 320 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.
a circle of radius r with center at (h,k) is
(x-h)^2 + (y-k)^2 = r^2
so, plug in your values.
radius is 160 ft
so, the center is at (0,9+160)
x^2 + (y-169)^2 = 160^2
To find the equation of the wheel, we need to determine the coordinates of the center and the radius of the wheel.
1. The center of the wheel can be determined by considering the diameter. Since the bottom of the wheel is 9 feet above the ground, the center of the wheel will be at a height of 9 + (320/2) = 169 feet above the origin.
2. The radius of the wheel is equal to half of the diameter, so it is 320/2 = 160 feet.
Therefore, the equation of the wheel with the origin at the ground directly below the center is:
(x - 0)^2 + (y - 169)^2 = 160^2
To find the equation of the Ferris wheel, we can start by understanding its shape. A Ferris wheel is a circle, and the center of the circle in this case is directly above the origin, since we are considering the origin as the ground directly below the center of the wheel.
The diameter of the Ferris wheel is given as 320 feet, which means the radius of the wheel is half of the diameter, or 320/2 = 160 feet.
The equation of a circle is given by the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius.
In this case, since the origin is the center of the Ferris wheel, the equation becomes x^2 + (y - k)^2 = r^2, where k is the distance between the center of the Ferris wheel and the ground.
Given that the bottom of the Ferris wheel is 9 feet above the ground, k = 9.
Plugging in the values, we get the equation:
x^2 + (y - 9)^2 = 160^2
Simplifying, the equation of the Ferris wheel is x^2 + (y - 9)^2 = 25600.
Therefore, the equation of the wheel is x^2 + (y - 9)^2 = 25600.