How do I find the focal distance?

Question

I’ll first put the question as it is on the assignment and then ask my question.

The following describes certain characteristics of a parabolic radio telescope that an astronomical observatory would like to build. It is presented in two positions: pointing upward and to the horizon. In this first position, the focus reaches a height of 18.4 m and the edge of the reflector is situated 14.5 m from the floor. In the second position, the bottom of the reflector is 2 m from the floor and the top is at 18 m. You should know that the efficiency of a radio telescope depends on the diameter of its reflector. The greater this diameter, the more the radio telescope is able to capture waves coming from remote regions of space. However, the capture quality also depends on the focal distance, which is the distance between the focus and the vertex. This distance must neither be too great nor too small relative to the diameter. In fact, what is important is the ratio of $\frac{focaldistance}{diameter}$, which ideally should be close to 0.5.

Then they go on asking,

a) Determine the value of this ratio for the radio telescope presented. b) To increase the value of the ratio without reducing the diameter of the reflector, what changes should be made to the shape of the parabola? State a conjecture justifying your point of view, using an example or reasoning.

So my question is:

How do I find the focal distance of the mentioned parabola? I tried to use the formulas in the book but I was not able to find it. I only found the diameter (18 m - 2 m = 16 m).

I have a feeling that the answer is right in front of me but I don’t see it.

Answer 1

You're right, from the second position information, we know that the diameter $d$ is

$ d = 18 - 2 = 16 m $

Now from the first position, and using the standard equation of a parabola,

$ y = \dfrac{ x^2 }{4 p } + b $

The focus of this parabola is at $(0, b + p ) $

Therefore,

$ 18.4 = p + b $

And from the edge information,

$ 14.5 = \dfrac{8^2}{4 p } + b $

Subtracting the two

$ 18.4 - 14.5 = 3.9 = p - \dfrac{16}{p} $

Hence,

$3.9 p = p^2 - 16 $

i.e.

$ p^2 - 3.9 p - 16 = 0 $

whose solutions are:

$ p = \dfrac{1}{2} ( 3.9 + \sqrt{ (3.9)^2 + 64 } ) = 6.4 m$

Hence, the required ratio is

$ \dfrac{p}{d} = \dfrac{6.4}{16} = 0.4 $

That was the solution of part(a).

For part (b), you want to keep the diameter $d$ unchanged, so you have to increase $p$. Ideally, we want $p$ to be equal to $8$.