Conics- Parabolas (Please help me and give as much detail as possible, I don't understand this)

Question

The reflector of a satellite dish is in the shape of a parabola with a diameter of 4 feet and a depth of 2 feet. To get the maximum reception we need to place the antenna at the focus.

a. Write the equation of the parabola of the cross section of the dish, placing the vertex of the parabola at the origin. Convert the equation into standard form, if necessary. What is the defining feature of the equation that tells us it is a parabola?

b. Describe the graph of the parabola. Find the vertex, directrix, and focus.

c. Use the endpoints of the latus rectum to find the focal width.

d. How far above the vertex should the receiving antenna be placed?

Answer 1

Given: Satellite dish is a parabola with diameter of 4.

Find: Focus.

Solution:

Visualize a half sphere, upside-down,

Sitting on the vertex.

That's our satellite dish.

Looking at it from the side, puts it in 2D

We have a parabola, vertex at (0,0)

and two points, on the edges: (-2,2) and (2,2)

The equation is: (4p)y = x^2

Use point (2,2) to solve for p

(4p)y = x^2

(4p)(2) = (2)^2

8p = 4

p = 1/2

Therefore, the focus is located at:

(x,y) = (0, 1/2)

The directrix is: y = -1/2

The focal width = 4p = 4(1/2) = 2

The focal width is the width of the parabola

going through the focus.

I like to think of the focus as the belly-button

and the focal-width like a belt, across the belly-button!