Question Statement

Playing team coaches use parabolic shaped antennas to listen to the conversation between the players in the ground. If the antenna has the cross section of 18 inches and depth of 4 inches, then where should the microphone be placed to hear the conversation?

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Background and Explanation

This problem utilizes the reflective property of parabolas: sound waves traveling parallel to the axis of symmetry reflect off the parabolic surface and converge at the focus. Conversely, sound emitted from the focus reflects off the surface as parallel rays. Therefore, to capture conversations from the field, the microphone must be positioned at the focus of the parabolic antenna.

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Solution

Let the equation of the parabola be:

$y^2=4ax$

Given that the cross section of the antenna is 18 inches and its depth is 4 inches.

Since the cross section (width) is 18 inches, the half-width is 9 inches. The depth represents the horizontal distance from the vertex to the rim. Thus, the coordinates of the points $A$ and $B$ on the rim are $(4,9)$ and $(4,-9)$ respectively.

Since point $\mathbf{A}(4,9)$ lies on the parabola, we substitute these coordinates into equation (i):

$(9{)}^2=4a(4)$

$81=16a$

$\Rightarrow a=\frac{81}{16}$

To hear the conversation, the microphone should be placed at the focus $(a,0)$.

Therefore, the microphone should be placed at:

$\left(\frac{81}{16},0\right)$

i.e., at a distance of $\frac{81}{16}$ inches (or $5.0625$ inches) from the vertex.

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Key Formulas or Methods Used

• Standard equation of right-opening parabola: $y^2=4ax$ where vertex is at origin and axis of symmetry is the x-axis
• Focus of parabola: For $y^2=4ax$, the focus is located at $(a,0)$
• Geometric interpretation: The focal length $a$ represents the distance from the vertex to the focus

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Summary of Steps

1. Set up the coordinate system: Place the vertex of the parabolic antenna at the origin $(0,0)$ with the parabola opening to the right, giving the standard equation $y^2=4ax$.
2. Identify boundary coordinates: With a cross section of 18 inches (total width) and depth of 4 inches, determine that the rim points are at $(4,9)$ and $(4,-9)$.
3. Solve for parameter $a$: Substitute the point $(4,9)$ into the parabola equation to get $81=16a$, yielding $a=\frac{81}{16}$.
4. Locate the focus: The focus is at $(a,0)=\left(\frac{81}{16},0\right)$.
5. State the final answer: The microphone should be placed $\frac{81}{16}$ inches (approximately $5.06$ inches) from the vertex along the central axis of the antenna.