- Analyze the symmetry of the given data points \((0, 27), (52.5, 12), (105, 7), (157.6, 12), (210, 27)\).
- Identify the horizontal axis of symmetry at \(x = 105\) using the midpoint formula.
- Recognize the corresponding height at the vertex, \(y = 7\), from the table.
- Conclude that the vertex of the parabola is \(\boxed{(105, 7)}\).
Explanation
- Problem Analysis and Data Inspection
The main cable of a suspension bridge forms a parabola, which can be represented by the vertex form equation \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. The provided data gives us coordinates \((x, y)\) representing the horizontal distance from the left support and the corresponding height of the cable: \((0, 27)\), \((52.5, 12)\), \((105, 7)\), \((157.6, 12)\), and \((210, 27)\). We observe that the data points show symmetry around a specific horizontal position. By examining the \(x\)-coordinates, we can identify this line of symmetry.
- Finding the x-coordinate of the Vertex (h)
A parabola defined by \(y = a(x - h)^2 + k\) is symmetric about the vertical line passing through its vertex, \(x = h\). We can determine \(h\) by looking at the symmetry of the given points. The points \((0, 27)\) and \((210, 27)\) are at the same height, suggesting the line of symmetry is at the midpoint of these two \(x\)-values:
\[h = \frac{0 + 210}{2} = 105.\]
Similarly, the points \((52.5, 12)\) and \((157.6, 12)\) are approximately symmetric around 105 (the average is \(\frac{52.5 + 157.6}{2} = 105.05\), which is very close to 105). Given that a bridge cable is expected to follow a precise parabolic curve, we can conclude that the axis of symmetry is exactly \(x = 105\).
- Determining the Vertex (h, k)
Since the vertex occurs at the axis of symmetry, its \(x\)-coordinate is \(h = 105\). Looking at the provided table, when the horizontal distance \(x\) is \(105\), the height \(y\) is \(7\). Thus, the vertex of the parabola is \((105, 7)\). The constant \(k\) represents the minimum or maximum height of the cable depending on the direction of the parabola; in this case, it represents the minimum height of the cable above the roadway.
Examples
Understanding the vertex of a parabolic cable is crucial for structural engineers when designing suspension bridges. By identifying the lowest point of the cable (the vertex), engineers can calculate the tension and load-bearing capacity needed to ensure the stability of the entire bridge. This geometric principle is applied in civil engineering to maintain structural integrity under various environmental conditions, ensuring that the roadway remains level and safe for traffic.