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 12 feet across at its opening and 2 feet deep at its​ center, at what position should the receiver be​ placed?

Answer 1

Answer:

18feet

Explanation:

We need to find the distance,c between the vertex and the focus.

For a vertical parabola:

$y=\frac{0.25}{c}(x-h)^2+k$

#If we place our parabola at the center, the expression changes to:

$y=\frac{0.25}{c}x^2$

The points of the parabola are given as (12,2), substitute in our equation to find c:

$y=\frac{1}{4c}x^2\\\\2=\frac{1}{4c}\times12^2\\\\c=12$

Hence, the receiver should be placed at 18ft

Answer 2

The position that the receiver should be placed is; 9 feet.

In this situation, one is to find the 'c' : that is, the distance between the vertex and the focus.

The equation for a vertical parabola is given as:

y = (¹/₄a)(x - h)² + k

where (h, k) are the coordinates of the center.

Now, if the parabola is at the center, then (h, k) = (0, 0)

 Thus; y = (¹/₄a)(x - 0)² + 0  

y = ¹/₄ax²

We are told that, the dish is 12 feet across at its opening and 2 feet deep at its​ center. thus;

Coordinate of given point is; (12,4)

Thus;  

y = ¹/₄ax²

4 =  ¹/₄a(12)²

4 =  ¹/₄a(144)

4 = 36a

a = 36/4

a = 9 ft

In conclusion, the position that the receiver should be placed is 9 feet.

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