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 4 feet deep at its​ center, at what position should the receiver be​ placed?

Answer 1

The receiver should be placed at 9 feet.

Explanation:

1. Let's say you are trying to find the unknown (a) which is the distance between the vertex and the focus.

To find the distance, the equation of the vertical parabola is used.

The equation is:

- y = (1/4a)(x-h)^2 + k

2. If we place our vertical parabola equation at the center our, equation becomes:

y = (1/4a)x^2

3. The problem gives us a point on the parabola: (12 , 4)

Changing unknowns would give that 'a':

4. y = (1/4a) x^2

4 = (1/4a) (12^2)

4 = (1/4a) 144

4 = (1/a) 36

4a = 36

a = 36/4

a = 9 feet

5. The answer is: 9 feet

Answer 2

c = 9 feet

Explanation:

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

Given that, the equation for a vertical parabola:

y = (1/4c)(x-h)^2 + k

Supposing we place our parabola at the center, our equation becomes:

y = (1/4c)x^2

.

The problem gives us a point on the parabola: (12,4)

Then insert it in and solve for 'c':

y = (1/4c)x^2

4 = (1/4c)12^2

4 = (1/4c)144

4 = (1/c)36

4c = 36

c = 36/4

c = 9 feet