A parabolic microphone has a cross-section in the shape of a parabola. The microphone is placed at the focus of the parabola. If the parabola is 20 inches wide and 5 inches deep, how far from the vertex should the microphone be placed?
Can someone please help.
To determine how far from the vertex the microphone should be placed, we need to use the properties of a parabola.
A parabola is defined by the equation y = ax^2, where (x, y) are the coordinates of any point on the parabola and 'a' is a constant that determines the shape of the parabola.
In this case, we know the width of the parabola is 20 inches, which means that the x-coordinate of the two points where the parabola intersects the x-axis (also known as the x-intercepts) are 10 inches and -10 inches.
The vertex of the parabola lies halfway between the x-intercepts, so the x-coordinate of the vertex is (10 + (-10)) / 2 = 0 inches.
Now, we need to find the y-coordinate of the vertex, which corresponds to the depth of the parabola.
Since the depth of the parabola is given as 5 inches, the y-coordinate of the vertex is -5 inches.
Therefore, the vertex of the parabola is located at (0, -5).
The microphone should be placed at the focus of the parabola, which is located at a distance equal to the y-coordinate of the vertex from the vertex itself.
In this case, the microphone should be placed 5 inches away from the vertex.
Note: In a parabolic microphone, the focus is the point where sound waves originating from different angles converge, allowing the microphone to capture sounds more effectively. Placing the microphone at the focus ensures that sound waves parallel to the axis of the parabola will reflect off the dish and converge at the microphone.