1. A Parabolic Antenna.
A parabolic antenna has the shape of a paraboloid of revolution: it is created by rotating part of a parabola around its axis. By the reflection property, all signals arriving parallel to the axis are concentrated at the focus after reflecting off the dish, so that is where the receiver is mounted. The axial cross-section of the dish is described by two measurements: the diameter \(d\) of the dish and its depth \(h\text{.}\) Placing the vertex at the origin with the dish opening upward, the cross-section is the parabola \(x^2 = 4py\text{,}\) and the rim passes through the points \(\left(\pm\tfrac{d}{2},\, h\right)\text{.}\)
Consider a dish antenna with diameter \(d = 120\) cm and depth \(h = 20.3\) cm, suitable for the amateur radio band at \(5.76\) GHz.
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A parabola opening upward with vertex at the origin. The rim points at (minus d over 2, h) and (d over 2, h) are joined by a horizontal double arrow labeled d. A vertical double arrow labeled h shows the depth from the rim down to the axis level. The focus F(0,p) lies on the y axis above the rim, and two segments run from the focus to the rim points, enclosing the opening angle two phi.
(a)
Determine the optimal location for the receiver, that is, find the distance from the vertex of the dish to the focus.
(b)
Find the quadratic function \(y = f(x)\) (explicit form) that describes the curvature of the dish, and graph it (for example, in GeoGebra or Desmos).
(c)
The opening angle \(2\varphi\) of the dish is the angle at which the two edges of the rim are seen from the focus. Compute it.
