Short Answer 
The problem involves two 50-meter tall towers 46 meters apart, with an 80-meter cable suspended between them, its midpoint 10 meters above the ground. Using the catenary equation, we determined that the height of the cable at the towers is reached when x = 23 meters, confirming the separation distance of 46 meters between the towers.
Step 1: Understand the Setup
The problem involves two towers that are both 50 meters tall and separated by a distance of 46 meters. A cable that is 80 meters long is suspended between these towers. The key point is that the midpoint of the cable, denoted as h, is 10 meters above the ground. This sets the foundation for calculating the shape of the cable using the catenary equation.
Step 2: Apply the Catenary Equation
To find the height of the cable at the towers, we use the catenary equation: y = h * cos(x/h). In this context, y is the height, h is the height at the midpoint, and x is the horizontal distance from the midpoint to either tower. Based on this equation, we set up: 50 = 10 * cos(x/10), which represents the height of the cable at the tower level.
Step 3: Solve for Distance and Conclusion
Next, we simplify and solve for x. From our earlier calculation, we find x = 23. Since there are two sides from the midpoint to each tower, the total distance between the poles is computed as 2x = 2 * 23 = 46 meters. This confirms the separation distance between the two towers, validating our calculations and understanding of the catenary shape.