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March 26, 2017

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An auditorium has a curved back wall. A horizontal cross section of the wall is in the shape of a parabola. The wall is 80 feet wide and the vertex of the parabolic cross section is 50 feet from the stage. Where in the auditorium should you sit for the best sound quality?

  • algebra 2 - ,

    The reason you are not getting answers is probably because the question is not very clear.

    I assume the following:
    1. The curved back wall is at the back of the auditorium, facing the stage.
    2. There is no side wall, so the curved back wall joins the stage near the front at 40' on each side of the centre stage.
    3. The "best sound quality" is at the focus of the parabola where we can hear everything going on on stage.

    We take the vertex of the parabola as the origin, and where it joins the stage as (±40,50).

    Assuming the standard equation of the parabola is y=x²/(4p), then we get
    p=x²/(4y)=40²/(4*50)=8.
    Since p is the distance of the focus from the vertex, then the focus of the parabola is at (0,8), or 8 feet from the vertex.

    At this position, all sound from the full width of the stage will be reflected by the backwall to the focus.

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