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 40 inches wide and 20 inches deep, how far from the vertex should the microphone be placed?
The vertex of the parabola is located midway between its focus and directrix, and the directrix is located 20 inches from the vertex. Therefore, the focus is also located 20 inches from the vertex.
Using the formula for the distance between a point on the parabola and its focus (p), we can solve for p:
p = (1/4)w = (1/4)(40 inches) = 10 inches
Therefore, the microphone should be placed 20 + 10 = 30 inches from the vertex of the parabolic dish.
To determine how far from the vertex the microphone should be placed in a parabolic microphone, we need to consider the properties of a parabola.
In a parabolic dish, the microphone is placed at the focus of the parabola. The focus of a parabola is located at a distance of one-fourth the width of the parabola.
Given that the width of the parabola is 40 inches, the distance from the vertex to the focus (where the microphone is placed) would be:
Distance from vertex to focus = 1/4 * width of the parabola
Distance from vertex to focus = 1/4 * 40 inches
Distance from vertex to focus = 10 inches
Therefore, the microphone should be placed 10 inches away from the vertex of the parabola.