Question 1170522
Let's break down this problem step by step.

**1. Set up a Coordinate System**

* Place the vertex of the parabola (the lowest point of the cable) at the origin (0, 30).
* The towers are 400 feet apart, so they are located at (-200, 150) and (200, 150).

**2. Find the Equation of the Parabola**

* The general equation of a parabola with a vertical axis of symmetry and vertex (h, k) is:
   (x - h)^2 = 4p(y - k)
* In our case, the vertex is (0, 30), so the equation becomes:
   x^2 = 4p(y - 30)

* We know the towers are at (200, 150). Plug these coordinates into the equation to find 'p':
   (200)^2 = 4p(150 - 30)
   40000 = 4p(120)
   40000 = 480p
   p = 40000 / 480 = 83.333... = 250/3

* Now, plug 'p' back into the equation:
   x^2 = 4(250/3)(y - 30)
   x^2 = (1000/3)(y - 30)

**3. Find the Height 50 Feet from a Tower**

* We want to find the height of the cable 50 feet horizontally from either tower. Since the towers are at x = ±200, we're looking for the height at x = 200 - 50 = 150 and x = -200 + 50 = -150. Due to symmetry, the height will be the same.
* Plug x = 150 (or x = -150) into the parabola equation:
   (150)^2 = (1000/3)(y - 30)
   22500 = (1000/3)(y - 30)
   22500 * (3/1000) = y - 30
   67.5 = y - 30
   y = 67.5 + 30
   y = 97.5

**4. Answer**

* The cable is 97.5 feet high 50 feet away (horizontally) from either tower.

**Final Answer:** 97.5 feet