16. A cable of a suspension bridge hangs in the form of a parabola, the Supporting towers of the cable being 300 meters apart.
over the supporting towers at a height of 60 meters above the roadwayand the lowest point of the cable is 5 meters above the roadway. Find
the lengths of the vertical supporting rods from the cable to the road. way at intervals of 50 meters from the center of the bridge to a supporting tower.
see your previous post. Was something unclear? If so, show how far you got.
6.67
To find the lengths of the vertical supporting rods from the cable to the roadway, we need to calculate the heights of the cable at specific intervals from the center of the bridge to a supporting tower. Let's break down the solution into steps:
Step 1: Determine the equation of the parabolic cable.
Since the cable hangs in the form of a parabola, we need to find an equation that represents its shape. We know the lowest point of the cable is 5 meters above the roadway, so the vertex of the parabola is at (0, 5). We also know that the supporting towers of the cable are 300 meters apart, which means the width of the parabola is 300 meters. Therefore, the equation of the parabola can be written as:
y = a(x - 0)(x - 300)
where 'a' is a constant that determines the shape of the parabola.
Step 2: Find the value of 'a'.
We can find the value of 'a' using the given information that the cable is at a height of 60 meters above the roadway over the supporting towers. Plugging in the coordinates for one of the towers, (150, 60), into the equation, we get:
60 = a(150 - 0)(150 - 300)
Simplifying, we get:
60 = a(-150)(-150)
60 = 22500a
Dividing both sides by 22500, we get:
a = 60/22500
a = 0.0026667
Therefore, the equation of the parabolic cable is:
y = 0.0026667x(x - 300)
Step 3: Calculate the heights of the cable at 50-meter intervals.
Starting from the center of the bridge (x = 0), we can calculate the height of the cable at 50-meter intervals until we reach one of the supporting towers (x = 150).
For x = 0:
y = 0.0026667(0)(0 - 300)
y = 0(0)(-300)
y = 0
The height of the cable at the center (x = 0) is 0 meters.
For x = 50:
y = 0.0026667(50)(50 - 300)
y = 0.0026667(50)(-250)
y = -3.33335
The height of the cable at 50 meters from the center is -3.33335 meters.
Continue this process at intervals of 50 meters until we reach x = 150, which is one of the supporting towers.
Step 4: Calculate the lengths of the vertical supporting rods.
To find the lengths of the vertical supporting rods, we need to subtract the height of the cable at each interval from the roadway height.
At x = 0, the height of the cable is 0 meters, and the roadway height is 5 meters. Therefore, the length of the vertical supporting rod is 5 - 0 = 5 meters.
At x = 50, the height of the cable is -3.33335 meters, and the roadway height is 5 meters. Therefore, the length of the vertical supporting rod is 5 - (-3.33335) = 8.33335 meters.
Repeat this calculation for each interval until reaching x = 150.