Question

The cross section of a parabolic reflector has a vertical axis of symmetry with its vertex at $\left(0,0\right)$ . The focus of the reflector is 6 feet above the vertex. The reflector extends 5.5 feet to either side of the vertex. What is the depth of the reflector? Round your answer to the nearest hundredth.

Answer 1

Answer:

The depth of the reflector is 700 feet.

Step-by-step explanation:

The cross section of a parabolic reflector is just a parabola (see the figure below). Because it has a vertical axis of symmetry and its vertex is at (0,0) the equation of the parabola is:

$y=4px^{2}\,\,(1)$

With p the distance to the focus (p=6ft), the equation for our particular case is:

$y=4(6)x^{2}\,\,(2)$

$y=24x^{2}\,\,(3)$

Note that because the reflector extends 5.5 feet to either side of the vertex, the extreme sides of the parabola are on the curve so they satisfy our parabola equation (3). Let’s concentrate on the right extreme of our parabola with x-position 5.5 ft using this number on (3) equation we can find the respective y-position

$y=24(5.5)^{2}\,\,\simeq\mathbf{700\,ft}(4)$

and that correspond to the depth of the parabolic reflector.