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.

Accepted Solution

Answer:The depth of the reflector is 0.84 feetStep-by-step explanation:(See the figure below)The equation of a parabola centered at the origin with an axis of symmetry on y-axis is:[tex] x^{2}=4py [/tex] (1)With p the distance from the origin to the focus using p=6, the equation (1) of the parabola becomes:[tex] x^{2}=4(6)y=24y [/tex] (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:[tex](4.5)^{2}=24y [/tex], solving for y[tex]y=\frac{(4.5)^{2}}{24}\approx0.84\,ft [/tex]