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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

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MrRoyal

Answer:

[tex]x^2 = 2y[/tex] --- equation

[tex](x,y) = (0,\frac{1}{2})[/tex] --- focus

[tex]y = -\frac{1}{2}[/tex] --- directrix

[tex]Width = 2[/tex] ---- focal width

Step-by-step explanation:

Given

[tex]depth = 2[/tex]

[tex]diameter = 4[/tex]

Required

The equation of parabola

The depth represents the y-axis. So:

[tex]y = 2[/tex]

The diameter represents how the parabola is evenly distributed across the x-axis.

We have:

[tex]diameter = 4[/tex]

-2 to 2 is 4 units.

So:

[tex]x = [-2,2][/tex]

So, the coordinates of the parabola is:

[tex](-2,2)\ and\ (2,2)[/tex]

The equation of the parabola is calculated using:

[tex]x^2 = 4py[/tex]

Substitute (-2,2) for (x,y)

[tex](-2)^2 = 4p*2[/tex]

[tex]4 = 8p[/tex]

Divide by 8

[tex]p = \frac{4}{8}[/tex]

[tex]p = \frac{1}{2}[/tex]

So, the equation is:

[tex]x^2 = 4py[/tex]

[tex]x^2 = 4 * \frac{1}{2} * y[/tex]

[tex]x^2 = 2y[/tex]

The defining features

(a) Focus

The focus is located at:

[tex](x,y) = (0,p)[/tex]

[tex](x,y) = (0,\frac{1}{2})[/tex]

(b) Directrix (y)

[tex]y = -p[/tex]

[tex]y = -\frac{1}{2}[/tex]

(c) Focal width

[tex]Width = 4p[/tex]

[tex]Width = 4*\frac{1}{2}[/tex]

[tex]Width = 2[/tex]

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