To find the appropriate distance from the bottom of the parabolic microphone dish to place the microphone, we can use the properties of a parabola. The formula for a parabola that opens upward is given in the standard form \( y = ax^2 \).
In this case, we know that the dish is 16 inches wide (which indicates the diameter) and 4 inches deep. The focal point of a parabola is located at a distance of \( \frac{1}{4p} \) from the vertex, where \( p \) is the distance from the vertex to the focus.
Given the dimensions:
- The width of the dish is 16 inches, which means the radius is 8 inches (since the diameter is twice the radius).
- The depth of the dish is 4 inches.
We can assume the vertex of the parabola is at the origin (0,0) and the focus is at (0, p) along the y-axis.
Since the vertex is at (0,0) and the bottom of the dish is at (0, -4), we can determine the equation by the following setup:
- When \( x = 8 \) (the edge of the dish), \( y = -4 \).
So we plug these values into the parabolic equation we set up earlier. Rearranging the equation gives us the relation, allowing us to isolate \( p \).
Now, let's figure out the distance from the vertex (the bottom of the trough of the dish) to the microphone, which is at the focus of the parabola. For a parabola defined by the equation \( y = \frac{x^2}{4p} \), it is known that the focus is located at \( (0, p) \).
- The maximum point of the parabola in height is therefore defined at \( y = -4 \) at \( x = 8 \).
- Set the standard parabola equation: \[ -4 = \frac{8^2}{4p} \] Simplifying this gives: \[ -4 = \frac{64}{4p} \implies -4 = \frac{16}{p} \implies p = \frac{16}{-4} = -4 \] Since we're measuring downward from the vertex, we actually want to keep in context the positive focus placement.
Thus the microphone should be placed \( 3.2 \) inches above the bottom of the dish (or 3.2 from the depth, which helps keep it close to the vertex) because the reflective property of a parabola shows that sound reaching any point along the curve will converge at the focus.
Therefore, the answer is:
3.2 in.