Question

The cross section of a parabolic reflector has a vertical axis of symmetry with its vertex at (0,0) . The focus of the reflector is 6 feet above the vertex. The reflector extends 4.5 feet to either side of the vertex. What is the depth of the reflector? Round your answer to the nearest hundredth.

Answer 1

The depth of the reflector is 0.84 feet

Explanation:

(See the figure below)

The equation of a parabola centered at the origin with an axis of symmetry on y-axis is:

`x^(2)=4py` (1)

With p the distance from the origin to the focus using p=6, the equation (1) of the parabola becomes:

`x^(2)=4(6)y=24y` (2)

Note that the point B is on the parabola, so this point should satisfy the parabola equation (2) that allow us to use the value x=4.5 to find the y value associated to it, that it is the depth (h) of the reflector:

`(4.5)^(2)=24y`, solving for y

`y=((4.5)^(2))/(24)\approx0.84\,ft`