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?

Answer 1

We can use the standard equation for a parabola with its vertex at the origin:

y = (1/4p) x^2

where p is the distance from the vertex to the focus. In this case, we know that the parabola is 40 inches wide, which means that the distance between the two points where the parabola intersects the x-axis (called the "latus rectum") is also 40 inches. That means:

4p = 40

p = 10

So the distance from the vertex (which is at the origin) to the focus is 10 inches. Since the microphone is placed at the focus, it should be 10 inches away from the vertex.

Answer 2

Let the equation for the parabola be
y =ax^2
We know that (20,20) is on the curve, so
a = 1/20 and our equation is
y = 1/20 x^2
Now recall that the parabola x^2 = 4py has its focus p units from its vertex. Our parabola is
x^2 = 20y, making p=5
so the focus is 5 units from the vertex.

Answer 3

So, the microphone should be placed 5 units away from the vertex, since it needs to be placed at the focus of the parabola.