The cable of a suspension bridge hangs in the form of a parabola. The distance between the two posts is 150 meters and the height of each post is 22 meters high. If the lowest point in the cable is 7 meters above the bridge, find the vertical distance to the cable from a point in the roadway 15 meters from the foot of a tower/post.

To solve this problem, we need to use the properties of a parabola.

1. First, we need to set up our coordinate system. Let's place the origin at the lowest point of the cable. The towers then are at points (-75, 15) and (75, 15) because the distance between the towers is 150 meters and the lowest point of the cable is 7 meters above the bridge, so the towers are 15 meters above the lowest point of the cable.

2. The equation of a parabola is y = ax^2 + bx + c. Since the parabola is symmetric with respect to the y-axis, the term with x (bx) will be zero. So, the equation simplifies to y = ax^2 + c.

3. We know that the parabola goes through the point (75, 15). Substituting these values into the equation gives us 15 = 75^2*a + c.

4. We also know that the parabola goes through the point (0, 0). Substituting these values into the equation gives us 0 = 0 + c. So, c = 0.

5. Substituting c = 0 into the equation from step 3 gives us 15 = 75^2*a. Solving for a gives us a = 15 / (75^2) = 0.004.

6. So, the equation of the parabola is y = 0.004x^2.

7. Finally, we need to find the vertical distance to the cable from a point in the roadway 15 meters from the foot of a tower. This point is at (-60, 0) or (60, 0). Substituting x = -60 or 60 into the equation gives us y = 0.004*60^2 = 14.4 meters.

So, the vertical distance to the cable from a point in the roadway 15 meters from the foot of a tower is 14.4 meters.