Okay, let's start solving this.
Our first step is to model the cables as a parabola. With the top of the towers being the highest points of the parabola, this would be the vertex. Also, because the cables touch the road level halfway between the towers, it means that the roots of the parabola are equidistant from the vertex. That is, if we put the vertex of the parabola at the origin of our coordinate system, the roots of the parabola would then be at (-1596/2, 0) and (1596/2, 0).
Next, we need to find the equation of the parabola. This equation is given as y = ax^2 + bx + c. But if we have the vertex at (h, k), we can write this in vertex form as given by a(x-h)^2 + k. Since our vertex is at (0, 142), substituting h and k in the equation, we have y = a(x - 0)^2 + 142, which simplifies to y = ax^2 + 142.
Let's now get the value of 'a'. We do this by fitting one of the roots of the parabola into the simplified equation above. Using the root (-1596/2, 0), we have 0 = a(-1596/2)^2 + 142. Solving this for 'a' gives us a = -142 / (1596/2)^2
Now, we want to find the height of the cables 250 feet from either end. We do this by substituting x = 250 or x = -250 in our equation y = ax^2 + 142. Thus, the height of the cable at x = 250 is given as a * 250^2 + 142.
Performing these calculations, we find that the height of the cables above the road 250 feet from either end of the bridge is approximately 128 feet to the nearest foot.