A cable of a suspension bridge hangs in the form of a parabola, the supporting towers of the cable being 300 meters apart. The cable passes over the supporting towers at a height of 60 meters above the roadway and the lowest point of the cable is 5 m above the roadway. Find the lengths of the vertical supporting rods from the cable to the roadway at intervals 50 meters from the center of the bridge to a supporting tower.
You have the parabola symmetric to the y-axis
y = ax^2+b
y(0) = 5
y(150) = 60
Now you can find a and b, and thus y(50)
To find the lengths of the vertical supporting rods from the cable to the roadway at intervals of 50 meters from the center of the bridge to a supporting tower, we can use the equation of a parabola.
Let's assume the center of the bridge is the origin of the coordinate system, and the positive x-axis extends towards the right. The two supporting towers will have x-coordinates of -150 and 150, respectively.
Now, let's denote the equation of the parabola as y = ax^2 + bx + c, where y represents the height of the cable above the roadway and x represents the distance from the origin to the point on the cable.
We know that the cable passes over the supporting towers at a height of 60 meters above the roadway. This information gives us two points on the parabola: (-150, 60) and (150, 60). Plugging these points into the equation of the parabola, we get:
60 = a(-150)^2 + b(-150) + c
60 = a(150)^2 + b(150) + c
Simplifying these equations, we have:
22500a - 150b + c = 60 -- Equation 1
22500a + 150b + c = 60 -- Equation 2
Now, we also know that the lowest point of the cable is 5 meters above the roadway. This information gives us a third point on the parabola: (0, 5). Plugging it into the equation of the parabola, we get:
5 = a(0)^2 + b(0) + c
5 = c
So, c = 5.
Now, substituting this value into Equations 1 and 2, we have:
22500a - 150b + 5 = 60 -- Equation 3
22500a + 150b + 5 = 60 -- Equation 4
Subtracting Equation 3 from Equation 4, we can eliminate 'b':
300b = 0
This implies b = 0.
Now, substituting the value of b into Equation 3 or Equation 4, we can find 'a'. Let's use Equation 3:
22500a + 5 = 60
22500a = 60 - 5
22500a = 55
a = 55 / 22500
a = 0.0024444
So, we have found the values of a, b, and c, which are a = 0.0024444, b = 0, and c = 5.
Now, we can find the height of the cable at any given x-coordinate using the equation of the parabola, y = 0.0024444x^2 + 5.
To find the lengths of the vertical supporting rods at intervals of 50 meters from the center of the bridge to a supporting tower, we need to calculate the height of the cable at those x-coordinates.
Starting from the center of the bridge (x = 0), we can go in increments of 50 meters to find the heights of the cable at each point:
- At x = -50: y = 0.0024444(-50)^2 + 5 = 6.111 meters
- At x = -100: y = 0.0024444(-100)^2 + 5 = 27.777 meters
- At x = -150: y = 0.0024444(-150)^2 + 5 = 60 meters
So, the lengths of the vertical supporting rods (height of the cable above the roadway) at intervals of 50 meters from the center of the bridge to the supporting tower are 6.111 meters, 27.777 meters, and 60 meters.