a mirror with a parabolic cross section is used to collect sunlight on 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 write an equation for a parabola with a vertex at the origin, we can use the standard form equation of a parabola: \(y = ax^2\).
Since the parabola opens upward, the coefficient \(a\) must be positive.
Given that the pipe is located 2 inches from the vertex of the mirror, we know that the distance from the focus to the vertex (which is also the distance from the focus to the directrix) is 2 inches.
In a parabolic geometry, the distance from the focus to the vertex is equal to \(1/(4a)\). Therefore, we have:
\(\frac{1}{4a} = 2\)
Simplifying this equation, we can solve for \(a\):
\(a = \frac{1}{8}\)
Therefore, the equation of the parabola is:
\(y = \frac{1}{8}x^2\)