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The main cable of a suspension bridge is modeled by the equation $y=0.0048(x-50)^2+6$. At what horizontal distances from the left support is the cable 10 ft above the roadway?



$y=$ height in feet of cable above the roadway

$x=$ horizontal distance in feet from left support



Choose the equation one must solve to answer the question:


A$0=0.0048(x-50)^2+6$
B$y=0.0048(10-50)^2+6$
C$10=0.0048(x-50)^2+6$

Verified Answer :

  • The height of the cable is given by the function \(y = 0.0048(x - 50)^2 + 6\).
  • To find the distance at a height of 10 feet, we set \(y = 10\).
  • Substituting this into the function results in the equation \(10 = 0.0048(x - 50)^2 + 6\).
  • The correct choice is \(\boxed{c}\).

Explanation

  1. Problem Analysis

The cable of the suspension bridge is defined by the height function \(y = 0.0048(x - 50)^2 + 6\), where \(y\) represents the vertical height in feet and \(x\) represents the horizontal distance in feet from the left support. To find the horizontal distances where the cable is \(10\) feet above the roadway, we must determine the value of \(x\) when the height \(y\) is equal to \(10\).

  1. Setting up the Equation

To find the horizontal distance \(x\) where the cable is at a height of \(10\) feet, we set \(y\) equal to \(10\) in the given model. By substituting \(10\) for \(y\) in the expression \(y = 0.0048(x - 50)^2 + 6\), we obtain the equation:
\[10 = 0.0048(x - 50)^2 + 6\]
This equation allows us to isolate \(x\) and find the specific horizontal points requested. Comparing this to the given options, it directly matches option (c).

  1. Conclusion

The equation that models the scenario where the cable height is \(10\) feet is \(10 = 0.0048(x - 50)^2 + 6\). Therefore, the correct choice is c.

Examples

In engineering and architecture, modeling the shape of structures using algebraic equations is essential. For instance, the suspension bridge example shows how engineers use parabolic functions to ensure the cable is at the correct height relative to the roadway. Similar techniques are used to design arches, analyze the trajectory of projectiles, or calculate the placement of support pillars in large-scale construction projects. Understanding these relationships allows for the precise calculation of structural dimensions and safety requirements.

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