A radio telescope has a parabolic surface as shown below.

A parabola opening up with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 9 meters and its width from left to right is 12 meters.

If the telescope is 9 m deep and 12 m wide, how far is the focus from the vertex?

Question

A radio telescope has a parabolic surface as shown below.

A parabola opening up with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 9 meters and its width from left to right is 12 meters. 

If the telescope is 9 m deep and 12 m wide, how far is the focus from the vertex?

e.mccormick · Answer

On the graph, you need to find some coordinates and use them to find the equation of the parabola.

anonymous · Answer

None are given

e.mccormick · Answer

Are there any marks on the graph?

anonymous · Answer

No, just x and y labels

e.mccormick · Answer

Well, if it is 9m deep and 12m wide, you are given numbers that can be used to find points.

anonymous · Answer

ok

e.mccormick · Answer

|dw:1406762867714:dw|

anonymous · Answer

here

e.mccormick · Answer

Yes, so, what three points do you 100% know with that diagram?

anonymous · Answer

0,9
-12,9
12,9

e.mccormick · Answer

Well, if it is 12 wide then it is not + and - 12.  And the vertex is given, which is 0,9.

anonymous · Answer

+- 12 makes sense on the graph

e.mccormick · Answer

No.  It is 12 wide.  Not from -12 to +12.  The distance from -12 to +12 is 24 wide.

anonymous · Answer

6,9
-6,9

e.mccormick · Answer

Yes.  And since the vertex is at the origin, (0,0) is also on the graph.

anonymous · Answer

ok

e.mccormick · Answer

There is one other nice thing about the vertex being on the graph.  All parabolas have the form of $y=ax^2+bx+c$ where the c is a shift up or down.  Well, it is at 0,0 for the vertex, so it can't be shifted up or down.  So c=0 is a given.

In fact, this parabola is also not shifted to the left or right at all... again because the vertex is at (0,0).  If it is shifted left or right or up or down, the vertex would be somewhere else.

anonymous · Answer

ok

e.mccormick · Answer

Another formula for all up/down parabolas is:
$(x-h)^2=4p(y-k)$ where (h,k) is the vertex.  Since that is 0,0:

$(x-0)^2=4p(y-0)$ 

$x^2=4py$

e.mccormick · Answer

So the question becomes, what form do you need to write it in?  Do you really need p?  Or can you use 4p as a single number, say if the invers of 4p is A, then you could use the form of:

$Ax^2=y$

anonymous · Answer

I need the distance between the focus and vertex

e.mccormick · Answer

So at this point, you keed to know how you must write the answer.  Once you know that, you can use your given points to find the equation.  Once you have the equation it gives you everything else.  See, in the $(x-h)^2=4p(y-k)$ form:

h = horozontal shift
k = vertical shift
(h,k+p) = focus
y=k-p = directrix

e.mccormick · Answer

So p is the distance between the focus and the vertex, so you need p.

e.mccormick · Answer

So put in a point, say (6,9), into $x^2=4py$ and solve for p.

anonymous · Answer

36=4p(9)
36=36p
p=1

e.mccormick · Answer

Looks like it.  So the distance from the focus to the vertex is 1m.

anonymous · Answer

Thanks

e.mccormick · Answer

Oh, and in real life the would never use a dish shaped the way they dimensioned that one. See, the pictured graph is deceptive. A 1 to 1 scale of that one is: https://www.desmos.com/calculator/iwbajopmpx See, does not look like a radio dish at all. Hehe. But it works for teaching.