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? Explain your reasoning and show all work.
To determine where in the auditorium you should sit for the best sound quality, we need to analyze the characteristics of the parabolic curve formed by the back wall.
First, let's understand the general shape of a parabola. A parabolic curve is defined by the quadratic equation in the form y = ax^2 + bx + c, where "a" determines the "stretch" factor, "b" adjusts the horizontal position, and "c" determines the vertical position (also known as the vertex).
In our case, the vertex of the parabolic cross-section is given as 50 feet from the stage. As the vertex represents the point of maximum or minimum value of the parabola (depending on the value of "a"), this implies that sitting near the vertex can potentially yield the best sound quality.
Now, let's focus on the width of the auditorium, which is given as 80 feet. Since the width corresponds to the horizontal axis of the parabola, it is helpful to find the equation of the parabolic curve to determine the position where it intersects the width.
Since the vertex is given, we can use the vertex form of the quadratic equation: y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.
In our case, h = 50 and k is unknown. To find k, we need to determine the value of y at the width of the auditorium (80 feet). So, we have the following equation:
0 = a(80 - 50)^2 + k
Simplifying, we get:
0 = 900a + k
Now, it is essential to know one more property of a parabola: symmetry. A parabolic curve is symmetric with respect to its vertex. This means that if we find the point where the curve intersects the width on one side of the vertex, we can mirror it to the other side.
Therefore, to find the position where the parabolic curve intersects the width on one side of the vertex, we divide the width by two (40 feet in this case) and add it to the x-coordinate of the vertex:
x = 50 + (80/2)
x = 90
Now, we have the x-coordinate of the point where the parabolic curve intersects the width. To find the corresponding y-coordinate, we substitute this value into the equation we derived earlier:
0 = 900a + k
Substituting x = 90:
0 = 900a + k
Next, we substitute the known width value (80) into the equation:
0 = a(80 - 50)^2 + k
0 = 900a + k
Thus, by equating the two expressions for k, we can solve for a:
0 = 900a + k
900a = 900a + k
0 = k
Now, we have both k = 0 and a. Since k represents the vertical position, we can conclude that the parabolic curve intersects the width at a height of 0 feet. This means that the curve touches the width at its lowest point.
To summarize, the best position to sit for optimal sound quality in the auditorium would be near the vertex since the parabolic shape of the back wall suggests that the sound waves will be focused and reflected towards this point. Therefore, sitting closer to the vertex, which is 50 feet from the stage, would likely provide the best sound quality.