A Dish TV satellite dish is the shape of a paraboloid. The dish is 42 inches wide, and 10 inches deep. How many inches should the receiver be located from the vertex for optimal reception? (round to the nearest thousandth)
To find the optimal location for the receiver, we need to determine the focal point of the paraboloid dish.
The focal length of a paraboloid is given by the equation f = (d^2)/(16c), where d is the depth of the dish and c is the width of the dish.
Plugging in the values, we have f = (10^2) / (16*42) = 0.1488.
Therefore, the receiver should be located approximately 0.149 inches from the vertex for optimal reception.
To determine the optimal location for the receiver, we need to find the focal point of the paraboloid. The focal point is located at a distance equal to one-fourth of the depth of the dish from the vertex.
In this case, the depth of the dish is 10 inches. So, the focal point can be found by dividing the depth by 4:
Focal point = 10 / 4 = 2.5 inches
Therefore, the receiver should be located 2.5 inches from the vertex for optimal reception.
the cross-section is a parabola
Recall that the parabola
x^2 = 4py
has its vertex at (0,0)
and its focus at y = p
You want y(21) = 10
That will give you the value of p you need.