Answer:
144 ft above the vertex
Step-by-step explanation:
You want the location of the focus of a paraboloid that is 144 ft across and 9 ft deep.
Equation
The equation of a parent parabolic function is ...
f(x) = x²
The points (1, 1) and (-1, 1) are on this curve, indicating its width is 1-(-1) = 2 at a depth of 1 unit.
To make the width be 144, we need to expand it horizontally by a factor of 144/2 = 72. To make the depth be 9, we need to expand it vertically by a factor of 9. Applying these scale factors gives us the function ...
[tex]g(x)=9\cdot f\left(\dfrac{x}{72}\right) = \dfrac{9x^2}{72^2}=\dfrac{x^2}{8\cdot72}[/tex]
Focus
When the equation is written in the form ...
[tex]y=\dfrac{1}{4p}x^2[/tex]
the value of p is the distance from the vertex to the focus.
We have ...
[tex]g(x)=\dfrac{1}{8\cdot72}x^2=\dfrac{1}{4\cdot144}x^2[/tex]
This shows us p = 144.
The receiver should be located at the focus, 144 feet from the vertex.
