Answer:
On the axis of symmetry, 64 ft from the vertex.
Step-by-step explanation:
You want the location of the focus of a paraboloid that is 64 ft across and 4 ft deep.
Equation
We know the equation y = x² has its vertex at (0, 0) and goes through the points (±1, 1). That is, it has width 2 at y = 1.
Scaling
We can vertically scale this relation by a factor of 4 to make it go through the points (±1, 4), and we can horizontally scale by a factor of 32 to make it go through points (±32, 4). The resulting equation is ...
[tex]y=4\left(\dfrac{x}{32}\right)^2=\dfrac{x^2}{256}[/tex]
Focus
Comparing this to y = x²/(4p) we find ...
4p = 256
p = 64 . . . . . . . . . the distance from the vertex to the focus
The receiver should be placed 64 ft from the bottom of the dish on the axis of symmetry.
