# Optimal Receiver Placement in Parabolic Satellite Dishes: Calculating the Distance from Vertex to Focus

Parabolic satellite dishes are a common sight in many parts of the world, used for receiving signals from satellites orbiting the Earth. The shape of these dishes is no accident – it’s a paraboloid, a three-dimensional shape that’s perfect for focusing incoming signals onto a single point: the focus. The placement of the receiver at this focus is crucial for the optimal functioning of the dish. But how do we calculate the exact distance from the vertex (the deepest point of the dish) to the focus? Let’s delve into the mathematics behind this.

## Understanding the Paraboloid

A paraboloid is a special type of surface in three-dimensional space. It’s defined by a parabolic curve that’s been rotated around its axis of symmetry. In the case of a satellite dish, this curve is a cross-section of the dish, and the axis of symmetry is the line running from the vertex to the focus.

## The Mathematical Formula

The formula that describes a paraboloid is y = x²/4f, where x is the distance from the axis of symmetry, y is the height, and f is the distance from the vertex to the focus. This formula is derived from the geometric properties of parabolas and can be used to calculate the focus of a parabolic dish.

## Applying the Formula

Let’s apply this formula to our example. We know that the dish is 10 ft across and 4 ft deep. This means that at the widest point (x = 5 ft, since the distance is measured from the axis of symmetry), the height is y = 4 ft. Substituting these values into the formula, we get 4 = 5²/4f. Solving for f, we find that f = 6.25 ft.