In a factory, a parabolic mirror to be used in a searchlight was placed on the floor. It measured 50 centimeters tall and 90 centimeters wide. Find the equation of the parabola.
To find the equation of the parabola, we need to determine its vertex and the focal length.
The vertex of the parabola is at the center of the mirror, which is halfway between the top and bottom of the mirror. Since the mirror is 50 centimeters tall, the vertex is at (0, 25).
The focal length of a parabolic mirror is equal to half of the height of the mirror. So, the focal length is 25 centimeters.
Since the mirror is positioned horizontally, we have a parabola opening upwards. The general equation for a parabola opening upwards with vertex (h, k) and focal length p is:
(x - h)^2 = 4p(y - k)
In this case, h = 0, k = 25, and p = 25. So, the equation of the parabola is:
x^2 = 4(25)(y - 25)
x^2 = 100(y - 25)
x^2 = 100y - 2500