A satellite dish has a diameter of 32 inches and is 6 inches deep. If the antenna needs to be put at the focus of the parabola, where would the parabola be placed?
I will place the parabola as opening to the right, with its vertex at (0,0)
so the equation is
y^2 = 4px, where (p,0) is the focus
but (6,16) lies on it
16^2 = 4p(6)
p = 256/24 = 32/3
So it should be placed at (32/3 , 0)
To find the location of the parabola in relation to the satellite dish, you need to understand the basic properties of a parabola. A parabola is a symmetric curve with a vertex, focus, and directrix.
In this case, the satellite dish is in the shape of a parabola, and the antenna needs to be placed at the focus of the parabola.
To determine the position of the parabola, you first need to find the focal length. The focal length is the distance from the vertex of the parabola to the focus.
The formula to find the focal length of a parabola is given by: f = (d^2) / (16h), where d is the diameter of the dish and h is the depth of the dish.
In this case, the diameter of the dish is 32 inches, and the depth is 6 inches. Let's calculate the focal length:
f = (32^2) / (16 * 6)
f = 1024 / 96
f ≈ 10.67 inches
So, the focal length of the parabola is approximately 10.67 inches.
Now that you know the focal length, you can determine the position of the parabola. The focus of the parabola will be located at the focal length behind the vertex, which is at the deepest point of the dish.
In this case, since the dish is 6 inches deep, the focus of the parabola will be at a distance of 10.67 inches behind the vertex (deepest point) of the dish. You can measure this distance on the dish from the deepest point towards the back to place the antenna correctly.
Therefore, the parabola would be placed approximately 10.67 inches behind the deepest point of the dish to position the antenna at the focus of the parabola.