衛星碟的拋物線與焦點

Problem

A satellite dish has cross-section $y = \dfrac{x^2}{8}$ and is $2$ feet deep; find the diameter of the opening and the location of the focus.

Step 1: Use the depth to find the rim points

Since the dish is $2$ feet deep, the opening is at $y = 2$. Substituting into $y = \dfrac{x^2}{8}$ gives

$$2 = \dfrac{x^2}{8}$$

so

$$x^2 = 16$$

and therefore

$$x = \pm 4.$$

Step 2: Compute the opening diameter

The rim points are $4$ feet to the left and right of the center, so the diameter is

$$2 \cdot 4 = 8.$$

Step 3: Match the parabola to standard form

Compare

$$y = \dfrac{x^2}{8}$$

with the standard form

$$y = \dfrac{x^2}{4p}.$$

This gives

$$4p = 8$$

so

$$p = 2.$$

Step 4: Locate the focus

With the vertex at the bottom of the dish, the focus is $2$ feet above it, so the focus is at

$$(0,2).$$

Answer

The dish opening is $8$ feet wide, and the focus is at $(0,2)$.