Question

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 7 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

[AI-generated answer — Bot GPT 3.5]

To find the equation of the parabola, we need to determine the coordinates of the focus of the parabola. Since the pipe is located at the focus of the mirror and is 7 inches from the vertex, the focus is located 7 inches above the vertex along the axis of symmetry of the parabola.

The general equation of a parabola with a vertical axis of symmetry, opening upwards, and with its vertex at the origin is given by:

y = a(x-h)^2 + k

where (h,k) is the vertex and 'a' is a constant that determines how wide the parabola opens.

Since the parabola has a focus 7 inches above the vertex, the coordinates of the focus are (0, 7). Since the vertex is at the origin, the vertex is (0,0). By the definition of a parabola, the distance between the focus and the vertex is the same as the distance between the vertex and the directrix, and this distance is known as the focal length. Therefore, the focal length is 7 units.

We know that the equation of a parabola with vertical axis of symmetry is given by:

4a * focal length = 1

Substitute the values of the focal length and solve for 'a':

4a * 7 = 1
28a = 1
a = 1/28

h = 0
k = 0

Therefore, the equation of the parabola that models the cross section of the mirror is:

y = (1/28)x^2