A mirror with a parabolic cross section is used to collect sunlight on a pipe located at the focus of the mirror. The pipe is located 9 inches from the vertex of the mirror. Write an equation of the parabola that models the cross section of the mirror. Assume that the parabola opens upward.
Let's assume that the vertex of the parabolic cross-section is at the origin (0,0) of the coordinate plane.
A parabolic equation in vertex form is given by: y = a(x-h)^2 + k, where (h,k) represents the vertex and a determines the shape of the parabola.
Since the vertex of the cross-section is the origin (0,0), we have h = 0 and k = 0.
The distance from the vertex (0,0) to the focus is defined as p, which in this case is equal to 9 inches. Since the parabola opens upward, p will be a positive value.
The equation of a parabola in terms of its focus (p,0) is given by: x^2 = 4py.
Substituting the values, we have:
x^2 = 4p(0 - y)
x^2 = -4py
Since p = 9, the equation becomes:
x^2 = -4(9)y
x^2 = -36y
Therefore, the equation of the parabola that models the cross-section of the mirror is x^2 = -36y.