Question

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 64 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

Answer 1

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 ...

  $y=4\left(\dfrac{x}{32}\right)^2=\dfrac{x^2}{256}$

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.