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?
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.
To determine the best seating location for sound quality in the auditorium, we need to analyze the shape of the parabolic cross-section of the back wall. In a parabola, sound waves emanating from the stage will bounce off the curved wall and converge at a focal point.
Given that the cross-section of the wall is a parabola and that the width of the wall is 80 feet, we can deduce that the distance from the stage to the vertex of the parabolic cross-section is 50 feet. This vertex represents the focal point of the parabola.
To find the optimal seating location, we search for the point on the axis of symmetry that is closest to the focal point (vertex). The axis of symmetry of a parabola is a vertical line that passes through the vertex and divides the parabola into two equal halves.
In this case, since the width of the wall is 80 feet, the axis of symmetry will be the line passing through the vertex and perpendicular to the stage, dividing the wall into two equal halves of 40 feet each.
The best seating location for optimal sound quality will be along the axis of symmetry, at a distance of 40 feet from the stage. This seating location maximizes the time it takes for sound waves to travel from the stage to the wall and back to the focal point, creating a surround sound effect for the audience seated along the axis of symmetry.
Therefore, for the best sound quality, you should ideally sit at a distance of 40 feet from the stage along the axis of symmetry in the auditorium.