Question

A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single​ point, where the receiver is located. If the dish is 14 feet across at its opening and 2 feet deep at its​ center, at what position should the receiver be​ placed?

Answer 1

Answer:

The receiver must be placed at a distance of 6.125 ft from the center.

Step-by-step explanation:

The dish is of parabola shape with the length of 14 feet and 2 feet deep from the center.

A parabola is a curve which is a set of all points in the plane which are at equal distance away from a given point called focus and given line called directrix.

We have to find out the point at which all the waves comes to one point where the receiver must be placed is the focus of the parabola.

The above scenario is shown in the image which can be converted in mathematical graphical representation of the parabolic curve.

The extreme two end points becomes, (-7,2) and (7,2)

The general equation for such parabola is:

x² = 4ay

where, a is the distance from the center and the focus.

Since, (7,2) satisfy the equation so,

7² = 4a×2

a = 49 /8

Thus,

The receiver must be placed at a distance of 6.125 ft from the center.