A civil engineer designs a suspension bridge with a cable modeled by $y = rac{1}{200}x^2 + 5$, where $x$ is horizontal distance from the center in meters and $y$ is height above river level. If the towers are 80 meters apart, what is the vertical distance between the lowest point of the cable and the top of a tower?

["Title: How the Shape of a Suspension Bridge Cable Determines Tower Height: Solving for the Vertical Gap", "Meta Description:\nExplore how a suspension bridge’s cable, modeled by the parabolic equation $y = \frac{1}{200}x^2 + 5$, determines the height between the lowest point of the cable and the top of its towers—80 meters apart. Learn the science behind this critical civil engineering design.", "---", "### Understanding Suspension Bridge Cables with Parabolic Models", "Suspension bridges rely on graceful, mathematically precise cable shapes to efficiently distribute structural loads. While real cables form a catenary, many simplified models approximate the curve using parabolas—especially when horizontal forces are balanced. The given equation, $ y = \frac{1}{200}x^2 + 5 $, models a cable’s vertical profile, where $ y $ is the height above the river level and $ x $ measures horizontal distance from the bridge’s center.", "In this scenario, the towers are spaced 80 meters apart, meaning each tower stands 40 meters from the bridge’s center (half the total span). To find how tall the bridge towers must be, engineers calculate the vertical difference between the lowest point of the cable and the top of a tower—a key factor in ensuring both structural integrity and aesthetic appeal.", "---", "### Step 1: Identify the Vertex of the Parabola", "The given equation is in vertex form:\n$$\ny = \frac{1}{200}x^2 + 5\n$$\nThis is a parabola that opens upward, with its vertex (lowest point) at $ (0, 5) $. Since the cable’s height above the river increases symmetrically from the center, the lowest point of the suspended cable lies at a height of:", "$$\ny_{\ ext{lowest}} = 5 \ ext{ meters}\n$$", "---", "### Step 2: Compute Cable Height at the Tower Position", "The towers are located 40 meters horizontally from the center (half of 80 meters span). Substitute $ x = 40 $ into the equation:", "$$\ny_{\ ext{tower}} = \frac{1}{200}(40)^2 + 5 = \frac{1600}{200} + 5 = 8 + 5 = 13 \ ext{ meters}\n$$", "So, the cable rises to a height of 13 meters above river level at each tower base.", "---", "### Step 3: Calculate Vertical Distance from Lowest Cable Point to Tower Top", "The vertical distance between the cable’s lowest point and the top of the tower is:", "$$\n\ ext{Vertical distance} = y_{\ ext{tower}} - y_{\ ext{lowest}} = 13 - 5 = 8 \ ext{ meters}\n$$", "---", "### Conclusion: Engineering Precision in Bridge Design", "This straightforward calculation—derived from the parabolic cable equation—reveals that the towers stand 8 meters tall (or more precisely, the cable crown is 8 meters above the lowest point) to meet the bridge’s 80-meter span. This vertical clearance ensures safe clearance for navigation or weather flow while maintaining the cable’s structural efficiency.", "Civil engineers use such models not only for accuracy but also to balance cost, strength, and functionality. By modeling the cable’s shape mathematically, they precisely determine tower heights, ensuring iconic suspension bridges like the Golden Gate or modern marvels rise safely and gracefully across vast waterways.", "---", "Keywords: suspension bridge design, cable equation $ y = \frac{1}{200}x^2 + 5 $, vertical clearance tower, civil engineering bridge modeling, parabolic cable tallest point, bridge structural analysis, low point height cable modeling", "---", "Explore more about the physics and geometry of infrastructure at the intersection of mathematics and civil engineering."]