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We are given a suspension bridge where the cable hangs in the shape of a parabola. Here's the information we have:

- The towers supporting the cable are **400 feet apart** (this is the horizontal distance between them).
- The height of the cable at the towers is **150 feet**.
- The lowest point of the cable is at the midpoint of the bridge, which is **30 feet above the bridge**.

We are asked to find how high the cable is **150 feet horizontally** away from either tower.

### Steps to solve:

1. **Set up the parabola equation**:
   Since the cable is in the shape of a parabola, we can model it with an equation of the form:

   \[
   y = ax^2 + bx + c
   \]

   where \(y\) is the height of the cable at any horizontal distance \(x\), and \(a\), \(b\), and \(c\) are constants to be determined.

2. **Choose a coordinate system**:
   - Let the midpoint of the bridge be at the origin \((0, 30)\), since the lowest point is 30 feet above the bridge at the midpoint.
   - The towers are 400 feet apart, so they are at \(x = -200\) and \(x = 200\), with heights of 150 feet.

   Using this setup, we know:
   - At \(x = 0\), \(y = 30\),
   - At \(x = 200\), \(y = 150\),
   - At \(x = -200\), \(y = 150\).

3. **Find the equation of the parabola**:
   Substituting \(x = 0\), we get:
   \[
   30 = a(0)^2 + b(0) + c \implies c = 30
   \]
   So, the equation is:
   \[
   y = ax^2 + bx + 30
   \]

   Now use \(x = 200\) and \(y = 150\):
   \[
   150 = a(200)^2 + b(200) + 30
   \]
   Simplify:
   \[
   150 = 40000a + 200b + 30 \implies 120 = 40000a + 200b \implies 120 = 400(100a + b)
   \]
   This gives the equation:
   \[
   y

The cable of a suspension bridge hangs in the shape of a parabola. The towers supporting the cable are 400 ft apart and 150 ft high. If the cable, at its lowest, is 30 ft above the bridge at its midpoint, how high is the cable 150 ft away (horizontally) from either tower?

Learn how to solve a real-world problem involving the parabolic shape of a suspension bridge cable. The towers are 400 feet apart, and the cable is 150 feet high at the towers and 30 feet high at its midpoint. Calculate how high the cable is 150 feet away from either tower using quadratic equations. Suitable for Grades 10-12 students.

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