A satellite dish is shaped like a paraboloid of revolution. the signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 14ft across at its opening and 4ft deep at its center, at what position should the receiver be placed?
To find the position where the receiver should be placed, we need to determine the focal point of the paraboloid.
The general form of a parabola equation in standard form is y^2 = 4ax, where "a" is the focal length.
In this case, we are given that the depth of the dish at its center is 4ft, which means the vertex is at (0, 4). We also know that the opening of the dish is 14ft, so the distance from the vertex to the opening is half of that, or 7ft.
To find the focal length "a", we can use the formula a = (d^2) / (16h), where "d" is the distance from the vertex to the opening and "h" is the depth of the dish at its center.
Plugging in the values, we have:
a = (7^2) / (16 * 4)
a = 49 / 64
a ≈ 0.7656 ft
Now that we have the focal length, we can find the position where the receiver should be placed. The focal point of a paraboloid of revolution is located along the axis of symmetry, which is the y-axis in this case. So, the receiver should be placed at a distance of 0.7656 ft (approximately) along the y-axis from the vertex, in the negative direction.
Therefore, the position where the receiver should be placed is (0, 4 - 0.7656) or approximately (0, 3.2344) ft.