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Answer:
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).
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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