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 2 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.
To determine the equation of a parabola, we use the standard form of the equation for a parabola that opens upwards, which is given by:
y = ax^2 + bx + c.
However, a parabolic mirror that focuses light onto a point uses a simpler form of the parabolic equation, which is derived from the vertex form of the parabola:
y = a(x - h)^2 + k,
where (h, k) is the vertex of the parabola. Since the parabola focuses light onto a pipe located at the focus and the parabola opens upward, we will assume the vertex of the parabola is at the origin (0,0) for simplicity.
In the case of a parabolic mirror, the focus is located at (0, p), where p is the distance from the vertex to the focus along the axis of symmetry. Since the problem states that the focus is located 2 inches from the vertex, p = 2 inches.
The standard form of a parabola with its vertex at the origin and the axis of symmetry along the y-axis is:
y = 4px,
where p is the distance from the vertex to the focus.
However, since the parabola opens upwards in this case, we need to rotate the standard equation 90 degrees, which means our axis of symmetry is the x-axis and the parabola opens in the positive y-direction. The rotated equation then becomes:
x = 4py.
For our parabola with p = 2 inches, the equation becomes:
x = 4 * 2y,
x = 8y.
This is the equation of the parabola that has a focus 2 inches from the vertex and opens upward.