The reflecting dish of a parabolic microphone has a cross-section in the shape of a parabola. The microphone itself is placed on the focus of the parabola. If the parabola is 60 inches wide and 30 inches deep, how far from the vertex should the microphone be placed?
Using your dimensions of the parabola, let's say that it is of the form
y = ax^2
Since y(30)=30, a = 1/30
y = 1/30 x^2
Now recall that the parabola
x^2 = 4py
has focus at (0,p). So, 4p=30 and p = 7/2. So the focus should be 3.5 inches from the vertex.
To determine how far from the vertex the microphone should be placed, we need to find the distance between the vertex and the focus of the parabola.
The focus of a parabola is located a distance of "p" from the vertex, where "p" is the distance between the vertex and the focus. In this case, "p" is given by the formula p = width / 4.
Given that the width of the parabola is 60 inches, we can calculate "p" as follows:
p = 60 / 4 = 15 inches.
Therefore, the microphone should be placed 15 inches from the vertex of the parabola.
To find the distance from the vertex of the parabola to the microphone, we need to determine the focal length of the parabola. The focal length is the distance from the vertex of the parabola to its focus.
In a parabolic dish, the focus is located at a distance of one-fourth the width of the dish. Let's calculate the focal length.
Given:
Width of the parabola (diameter): 60 inches
Depth of the parabola: 30 inches
Since the width is given as the diameter, we need to find the radius of the parabolic dish, which is half of the width.
Radius = Width / 2
Radius = 60 / 2
Radius = 30 inches
The focal length of a parabolic dish is given by the equation:
Focal Length = Radius^2 / (4 * Depth)
Substituting the given values:
Focal Length = 30^2 / (4 * 30)
Focal Length = 900 / 120
Focal Length = 7.5 inches
Therefore, the microphone should be placed at a distance of 7.5 inches from the vertex of the parabola.