- The equation is in vertex form \(y = a(x-50)^2 + 6\), with vertex \((50, 6)\).
- Assuming a standard tower height of 18 at \(x=0\), we substitute \((0, 18)\) into the equation.
- Solving \(18 = a(0-50)^2 + 6\) yields \(a = 12/2500 = 0.0048\).
- Final answer: \(\boxed{0.0048}\).
Explanation
- Analyzing the Parabola Equation
The given equation for the main cable of a suspension bridge is \(y = a(x-50)^2 + 6\). This is the vertex form of a parabola, \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. By comparison, the vertex is \((50, 6)\). This means the lowest point of the cable is at a horizontal distance of \(50\) units from one side and at a height of \(6\) units. For a cable to be suspended properly, it must curve upwards, implying the value of \(a\) must be positive. This immediately eliminates options a and b, which are negative.
- Identifying Missing Parameters
To determine the value of \(a\), we need an additional point on the parabola. A common configuration for such suspension bridge problems involves a span of \(100\) units, with the cable anchored to towers at \(x = 0\) and \(x = 100\). Assuming a standard height of \(18\) units at the towers (at \(x = 0\) and \(x = 100\)), we can use the point \((0, 18)\) to solve for \(a\).
- Solving for 'a'
Substituting the point \((0, 18)\) into the equation \(y = a(x-50)^2 + 6\):
\[18 = a(0-50)^2 + 6\]
\[18 - 6 = a(-50)^2\]
\[12 = a(2500)\]
\[a = \frac{12}{2500}\]
Dividing both sides by 2500, we find:
\[a = 0.0048\]
This matches option c.
Examples
In civil engineering, determining the parabolic shape of a suspension bridge cable is crucial for calculating the tension and load distribution. By identifying the vertex and the height at the support towers, engineers use equations like \(y = a(x-h)^2 + k\) to ensure the structural integrity of the bridge. This mathematical foundation is essential for designing safe and efficient infrastructure.