Workers are designing a TV receiving dish in the shape of a paraboloid of a revolution, the factory sent them a paraboloid that is ten feet across at the opening and three feet deep. What is the best place the receiver can be placed for the best reception?
The "receiver", which is the microwave detector of the satellite signal, should be located at the focus of the paraboloid. The equation that generates the parabaloid is y = a x^2, where y = 3 when x = 5. Therefore a = 3/25 ft^-1
The focus of a parabola with formula y = a x^2 is at 1/4a, which is 25/12 ft in this case. That would be 25 inches from the bottom of the dish
if y=ax^2, then f=1/4a
at x=5, y=3, so
a=3/25 so f=25/12 feet up from the base.
To determine the best place to position the receiver for optimal reception in a paraboloid TV receiving dish, we need to consider the properties of the paraboloid and its focal point.
A paraboloid of revolution is a three-dimensional curved surface that can be formed by rotating a parabola around its axis. In this case, the paraboloid is described as being ten feet across at the opening and three feet deep. It is important to note that the term "ten feet across" likely refers to the diameter of the opening, meaning that the width of the opening is ten feet.
In a paraboloid of revolution, the focal point is located at the vertex of the parabola, above the opening of the dish. The focal point is where incoming rays of light or signals parallel to the axis will converge after reflecting off the surface of the paraboloid.
To determine the best place to position the receiver for the best reception, the workers should place it at the focal point of the paraboloid. Since the depth of the paraboloid is given as three feet, the focal point will be located three feet above the opening of the dish, at the vertex of the parabola.
By placing the receiver at the focal point, it will be in the position where it can collect the incoming signals most effectively. This is because the paraboloid is designed to reflect incoming signals towards the focal point, resulting in concentrated and focused signals at that location.
Therefore, placing the receiver at the focal point of the paraboloid will yield the best reception in a TV receiving dish of this shape.