To determine the correct placement of the microphone in a parabolic dish, we first need to find the focal point of the parabola that describes the dish.
The equation of a parabola that opens upwards can be expressed in the form:
\[ y = \frac{1}{4p}x^2 \]
where \( p \) is the distance from the vertex to the focus (the focal length).
In this case, the width of the dish is 24 inches, so the radius is 12 inches (half of the width). The depth of the dish is 5 inches, which means that at \( x = 12 \) (the edge of the dish), the value of \( y \) (the depth) is 5 inches.
Setting up the equation with these values:
\[ y = \frac{1}{4p}x^2 \]
We can substitute \( x = 12 \) and \( y = 5 \):
\[ 5 = \frac{1}{4p}(12)^2 \] \[ 5 = \frac{144}{4p} \] \[ 5 = \frac{36}{p} \]
Now, we can solve for \( p \):
\[ p = \frac{36}{5} = 7.2 \text{ inches} \]
Thus, the microphone should be placed \( p = 7.2 \) inches from the bottom of the dish.
Therefore, the answer is:
7.2 in.