The main cable of a suspension bridge forms a parabola, described by the equation $y=a(x-h)^2+k$, where $y$ is the height in feet of the cable above the roadway, $x$ is the horizontal distance in feet from the left bridge support, $a$ is a constant, and ($h, k$) is the vertex of the parabola.
At a horizontal distance of 30 ft, the cable is 15 ft above the roadway. The lowest point of the cable is 6 ft above the roadway and is a horizontal distance of 90 ft from the left bridge support.
Which quadratic equation models the situation correctly?
Answer (2)
The vertex of the parabola is ( 90 , 6 ) , so the equation is y = a ( x − 90 ) 2 + 6 .
Substitute the point ( 30 , 15 ) into the equation to find a : 15 = a ( 30 − 90 ) 2 + 6 .
Solve for a : a = 400 1 .
The quadratic equation that models the situation is y = 400 1 ( x − 90 ) 2 + 6 .
Explanation
Finding the vertex We are given that the main cable of a suspension bridge forms a parabola described by the equation y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. We are also given that the lowest point of the cable is 6 ft above the roadway and is a horizontal distance of 90 ft from the left bridge support. This means the vertex of the parabola is at ( 90 , 6 ) , so h = 90 and k = 6 . Thus, the equation becomes y = a ( x − 90 ) 2 + 6 .
Using the given point We are also given that at a horizontal distance of 30 ft, the cable is 15 ft above the roadway. This means when x = 30 , y = 15 . We can use this information to find the value of a . Substituting these values into the equation, we get 15 = a ( 30 − 90 ) 2 + 6 .
Solving for a Now, we solve for a . We have 15 = a ( − 60 ) 2 + 6 , which simplifies to 15 = 3600 a + 6 . Subtracting 6 from both sides gives 9 = 3600 a . Dividing both sides by 3600 gives a = 3600 9 = 400 1 .
Final equation Therefore, the quadratic equation that models the situation is y = 400 1 ( x − 90 ) 2 + 6 .
Examples
Suspension bridges are designed using mathematical models to ensure stability and safety. The parabolic shape of the main cable is crucial for distributing the load evenly across the bridge. By understanding quadratic equations, engineers can accurately predict the cable's height at any point along the bridge, which is essential for construction and maintenance. For instance, knowing the exact height at specific intervals helps in installing vertical support cables and assessing the structural integrity of the bridge over time. This ensures the bridge can withstand various loads and environmental conditions, making it a safe and reliable infrastructure.
The quadratic equation that models the height of the cable of the suspension bridge is given by y = 400 1 ( x − 90 ) 2 + 6 . This is derived by determining the vertex and substituting a known point into the equation. Solving for the constant a leads us to find the exact equation of the parabola.
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