A parabolic microphone used on the sidelines of a professional football game uses a reflective dish 16 in. wide and 4 in . deep. How far from the bottom of the dish should the microphone be placed? (1 point) 3 in . 3.2 in . 3.8 in . 4 in .

To find out how far from the bottom of the dish to place the microphone, we can model the parabolic dish as a parabola in the standard form \(y = \frac{x^2}{4p}\), where \(p\) is the distance from the vertex (bottom of the dish) to the focus (where the microphone should be). Given that the dish is 16 inches wide, the parabola opens sideways and has a width of 16 inches. The depth of the dish (4 inches) helps us find the value of \(p\). Setting the vertex at the origin \((0,0)\) and the overall width of the parabola as 16 inches, we sketch it from \((-8, 0)\) to \((8, 0)\) at the width and at a height (depth) of 4 inches. At the edge, the parabola meets the line \(y = -4\) when \(x = 8\). Substituting these coordinates into the parabola equation gives: \[ -4 = \frac{8^2}{4p} \] This simplifies to: \[ -4 = \frac{64}{4p} \] From which it follows that: \[ -4p = 64 \quad \Rightarrow \quad p = -16 \] Now since the focus (where the microphone should be placed) is located at \(p\) units above the vertex, we take the positive value of \(p\) as 16 inches (since we consider distances without sign). Therefore, \(p = \frac{Width^2}{16} = 16 \div 4 = 4\). Thus, the microphone should be placed **4 inches** away from the bottom of the dish, which leads us to select the option: 4 inches.