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.

Answer 1

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.