Answer:
Explanation:
To determine the position where the receiver should be placed, we need to understand the properties of a parabolic shape and the focus of the parabola.
A parabola is a curved shape with a specific property: any ray of light that enters the parabola parallel to its axis of symmetry will be reflected by the parabolic surface and pass through a single point called the focus.
In this case, the satellite dish is shaped like a parabola, and the signals from the satellite strike the surface of the dish and are reflected to a single point, where the receiver should be placed.
Given that the dish is 16 feet across at its opening, we can consider this as the width of the parabolic shape. This width is also known as the diameter of the dish.
We are also given that the dish is 5 feet deep at its center. This depth represents the distance between the center of the dish and the vertex of the parabolic shape.
Since we are interested in finding the position where the receiver should be placed, we need to locate the focus of the parabola.
To find the position of the focus, we can use the formula for the focal length of a parabola, which is given by the equation:
f = (d^2) / (16h)
Where:
- f represents the focal length
- d represents the depth of the dish at its center
- h represents half of the width of the dish
In this case, d = 5 feet and h = 8 feet (half of the 16 feet width).
Substituting these values into the formula, we have:
f = (5^2) / (16 * 8) = 25 / 128 ≈ 0.1953 feet
Therefore, the focal length of the parabolic shape is approximately 0.1953 feet.
Since the focus is the point where the receiver should be placed, we can conclude that the receiver should be placed approximately 0.1953 feet from the center of the dish towards the opening.