Final Answer:
The height of one of the towers is 63 meters.
Step-by-step explanation:
Let's use the properties of a parabola to solve this problem. The equation of a parabola can be written in the form y = ax^2 + bx + c, where a, b and c are constants.
The cable of the suspension bridge follows the shape of a parabola with its vertex in the center of the bridge.
The towers are 168 meters apart, so the center of the bridge is halfway between the towers, which is 84 meters from each tower.
We are told that the cable (the parabola) is 28 meters high when it is 56 meters from the center of the bridge. This gives us a point on the parabola which we can use to find the equation.
Because the parabola is symmetrical, and the vertex is at the center of the bridge, we know that the axis of symmetry is vertical, and the vertex is at the origin, (0, 0), since there is no horizontal shift given in the problem.
Thus, the equation simplifies to y = ax^2, because the line of symmetry of the parabola is the y-axis (b = 0) and the vertex at the origin means c = 0.
Now we know that when x = 56 (56 meters from the center of the bridge), y = 28 (the height of the cable). Plugging these values into the equation gives us:
28 = a(56)^2
To solve for a, we simplify the right side and then divide by it:
28 = a × 3136
a = 28/3136 = 1/112
So, the equation of the parabola is:
y = 1/112 × x^2
Now we want to find the height of the cable when it is directly above the point where a tower supports it (84 meters from the center), so we substitute x = 84 into the equation:
y = 1/112 × 84)^2
Solving for y:
y = 1/112 × 7056
y = 63
Therefore, the height of the cable at the point where the tower supports it is 63 meters. And since this is the height of one of the towers as well, the solution to the question is that the height of one of the towers is 63 meters.