A satellite dish is the shape of a paraboloid. The dish is 30 inches wide, and 10 inches deep. How many inches should
the receiver be located from the vertex for optimal reception?
(round to the nearest thousandth)
Receiver •

The distance that the receiver should be located from the vertex to receive optimal reception is;

5.625 inches above the vertex

We are given;

Width of the satellite dish = 30 inches

Depth of the satellite dish = 10 inches

Vertex of parabola is at (0, 0)

If we imagine a parabola that has its vertex located at the origin, the symmetry of the parabola will make us know that the depth  is the height which in this case is 10 inches while the edge is 15 inches (1/2 of 30 inches) to both right and left sides of the axis of symmetry.

Thus, we can say that this parabola passes through the point (15, 10).

Now, the conic form of a parabola with its vertex at the origin is given by the formula; y = (1/4c)x²

where c is the distance between the focus and the vertex.

Plugging in the relevant values gives;

10 =  (1/4c)(15)²

4c = 225/10

4c = 22.5

c = 22.5/4

c = 5.625

Thus, the receiver should be located at 5.625 inches above the vertex so for optimal reception.

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The receiver should be located at (0, 5.625)

Step-by-step explanation:

The given parameters are;

The width of the satellite dish = 30 inches

The depth of the dish = 10 inches

The receiver should be located at the focus of the parabola

Where the vertex of a parabola is (0, 0)

The equation of an upward or downward facing parabola is x² = 4·p·y

The focus of the parabola = (0, p)

The directrix is given by y = -p

The axis is the x-axis

Therefore, given that the width = 30 inches and the depth = 30 inches, we have;

The points (-15, 10), (15, 10), and (0, 0) (which is the vertex) on the parabola

Substituting, the coordinates into the equation of the parabola gives;

(-15)² = 4 × p × 10

p = (-15)²/(4 × 10) = 5.625

The coordinate of the focus = (0, 5.625)

Therefore, the receiver should be located at (0, 5.625).

We use the property of combination therefore the answer is 22c2= (22×21)÷(2×1) = 231 ways