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The problem describes the cables of the Golden Gate Bridge as following a parabolic shape, with towers that are 1280 meters apart and 160 meters tall. The cable touches the road at the midpoint between the towers. The goal is to find the height of the cable 200 meters away from one of the towers.

Let's set up a coordinate system with the origin at the midpoint of the bridge, where the cable touches the road. This means the equation of the parabola will be centered at the origin \((0, 0)\), and the equation of a parabola in standard form is \( y = ax^2 \).

### Step-by-step process:

1. **Determine the equation of the parabola:**
   - The total distance between the towers is 1280 meters, so each tower is 640 meters from the midpoint.
   - The height of the towers is 160 meters, so the point \((640, 160)\) lies on the parabola.

   Using the point \((640, 160)\) in the equation \( y = ax^2 \):
   \[
   160 = a(640^2)
   \]
   Solving for \(a\):
   \[
   a = \frac{160}{640^2} = \frac{160}{409600} = \frac{1}{2560}
   \]

2. **Find the height at 200 meters from a tower:**
   The distance from the midpoint to the point we are interested in is \(640 - 200 = 440\) meters.
   Now, substitute \(x = 440\) into the equation \(y = \frac{1}{2560}x^2\):
   \[
   y = \frac{1}{2560} \times 440^2
   \]

Let me calculate this next.The height of the cable 200 meters away from one of the towers is approximately 76 meters, when rounded to the nearest meter.

Would you like any additional details or further explanations on this? Here are 5 related questions you might find interesting:

1. How is the equation of a parabola derived from given points in general?
2. Can the equation for the parabola of the bridge be expressed in other forms?
3. How would the height change if the towers were placed further apart?
4. What are other real-world examples of structures modeled by parabolas?
5. How would you calculate the length of the cable between the two towers?

**Tip:** When working with parabolas in real-world problems, always choose a convenient coordinate system to simplify calculations (e.g., placing the origin at the vertex).

The towers of the Golden Gate Bridge connecting San Francisco to Marin County are 1280 meters apart and rise 160 meters above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the two towers. What is the height of the cable 200 meters from a tower? Round to the nearest meter.

Learn how to calculate the height of the parabolic cable of the Golden Gate Bridge 200 meters away from a tower. This problem involves concepts of parabolas, quadratic functions, and properties of quadratic equations, with a step-by-step solution for students in Grades 10-12.

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