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A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 96 feet across at its opening and 8 feet deep at its center, where should the receiver be placed?

Respuesta :

jmonterrozar

Answer:

72 feet

Step-by-step explanation:

We have that the vertex at the origin in an upward concave is:

y = a * x ^ 2, we solve for a:

a = y / (x ^ 2)

Thus:

The points from the origin (0,0) are (-48, 8) and (48, 8), we replace:

a = 8 / (48 ^ 2) = 1/288

Therefore the equation would be:

y = (1/288) * x ^ 2

288 * y = x ^ 2

Now, be distance above the vertex to put the receiver, which would be the focus, we have:

4 * p = 288

we replace:

4 * p = 288

p = 288/4

p = 72

xero099

The receiver should be placed at [tex](x,y) = (0, 72)\,[ft][/tex].

We assume that the paraboloid is centered at the origin of a rectangular system of coordinates, then we have the following standard equation of the parabola is:

[tex]4\cdot p \cdot y = x^{2}[/tex] (1)

Where:

  • [tex]x[/tex] - Horizontal distance with respect to origin, in feet.
  • [tex]y[/tex] - Vertical distance with respect to origin, in feet.
  • [tex]p[/tex] - Distance between vertex and origin, in feet.

The coordinates of the vertex are expressed by [tex](x,y) = (0, p)[/tex].

If we know that [tex]x = 48[/tex] and [tex]y = 8[/tex], then the coordinates of the vertex are, respectively:

[tex]4\cdot p\cdot 8 = 48^{2}[/tex]

[tex]p = 72\,ft[/tex]

The receiver should be placed at [tex](x,y) = (0, 72)\,[ft][/tex].

To learn more on parabolae, we kindly invite to check this verified question: https://brainly.com/question/8495268