An engineer designs a cable of a suspension bridge that hangs in the form of a parabola, the

towers supporting the cable are 360 meters apart. The cable passes over the supporting towers at a height of 80 meters above the roadway and the lowest point of the cable is 10 meters above the roadway Find the lengths of the vertical supporting rods from the cable to the roadway at intervals 60 meters from the center of the bridge to a supporting tower?

Set up the parabola with its vertex at (0,10). The equation is

y = ax^2 + 10
Since the 80m towers are 180 meters from the center,
a*180^2 + 10 = 80, so a = 7/3240
Thus the equation is
y = 7/3240 x^2 + 10
Now just find y at x = ± 60,120

To find the lengths of the vertical supporting rods, we need to determine the equation of the parabola that represents the cable.

Let's assume the vertex of the parabola is at point (0, 10). Since the cable passes over the supporting towers at a height of 80 meters above the roadway, we can assume the length from the vertex to the towers is 80 meters.

Using these details, we can write the equation of the parabola in vertex form:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola. In our case, h = 0 and k = 10. The equation becomes:

y = ax^2 + 10

Now we need to find the value of 'a.' Since the cable passes over the supporting towers at a height of 80 meters above the roadway, the coordinates of the towers are (180, 80) and (-180, 80). We can substitute these values into the equation to find 'a.'

80 = a(180)^2 + 10
80 = a(32400) + 10
80 - 10 = 32400a
70 = 32400a
a = 70 / 32400
a ≈ 0.00216

Now we have the equation of the cable:

y = 0.00216x^2 + 10

To find the lengths of the vertical supporting rods at intervals 60 meters from the center of the bridge to a supporting tower, we can substitute specific x-values into the equation and find the corresponding y-values.

At x = 60:
y = 0.00216(60)^2 + 10
y ≈ 78.72

The length of the vertical supporting rod at x = 60 is approximately 78.72 meters.

At x = -60:
y = 0.00216(-60)^2 + 10
y ≈ 78.72

The length of the vertical supporting rod at x = -60 is also approximately 78.72 meters.

In summary, the lengths of the vertical supporting rods from the cable to the roadway at intervals 60 meters from the center of the bridge to a supporting tower are approximately 78.72 meters.

To find the lengths of the vertical supporting rods from the cable to the roadway at intervals of 60 meters from the center of the bridge to a supporting tower, we first need to understand the shape of the cable.

The cable is in the form of a parabola, passing over the supporting towers at a height of 80 meters above the roadway. The lowest point of the cable is 10 meters above the roadway. The distance between the supporting towers is 360 meters.

We can use the equation of a parabola to represent the cable. The general form of a parabola equation can be written as:

y = a(x - h)^2 + k

Where:
- (h, k) represents the vertex of the parabola
- a represents a constant that determines the shape and orientation of the parabola

Since the given parabola passes over the supporting towers, we know that the coordinates of the two towers are (180, 80) and (-180, 80). Therefore, the vertex of the parabola is at the midpoint between the two towers, which is (0, 80).

Now, let's find the value of a.
Since the lowest point of the cable is 10 meters above the roadway, we can substitute the coordinates of this point into the equation:
10 = a(0 - 0)^2 + 80
10 = 80
This equation is not satisfied, which means there is no parabola that passes through the given points.

Therefore, it seems that there might be some mistake in the given information. Please double-check the values and provide the correct details so that we can accurately calculate the lengths of the supporting rods.