A cross section of a light reflector is a parabola. The bulb is located at the focus. Suppose the bulb is located 5 in from the vertex of the reflector. Use this information to make the following determinations.
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Part 1
Model a cross section of the reflector by writing an equation of a parabola that opens upward and has its vertex at the origin.
To model a parabola that opens upward and has its vertex at the origin, we can use the standard form of the equation of a parabola:
y = ax^2
Since the parabola opens upward, the coefficient "a" should be positive. The vertex of the parabola is at the origin (0,0), so the equation becomes:
y = ax^2
Now, given that the bulb is located 5 inches from the vertex of the reflector, we know that the focus of the parabola is located at (0,5). The focus of a parabola is located a distance of "p" units from the vertex along the axis of symmetry.
For a parabola that opens upward, the focus is located above the vertex. Using the formula for the distance from the vertex to the focus of a parabola given by p = 1/(4a), we have:
5 = 1/(4a)
4a = 1/5
a = 1/20
Therefore, the equation of the parabola that models the reflector is:
y = (1/20)x^2