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.
To find the lengths of the vertical supporting rods from the cable to the roadway, we can utilize the equation of a parabola. Let's assume that the equation of the cable is given by y = ax^2 + bx + c, where the x-axis represents the distance from the center of the bridge to a supporting tower.
First, let's find the equation of the parabola using the given information. We know that the highest point of the cable is 60 meters above the roadway, which means that it passes through the point (0, 60). Similarly, the lowest point of the cable is 5 meters above the roadway, which means it passes through the point (150, 5) since the supporting towers are 300 meters apart.
Substituting these points into the equation, we get the following system of equations:
1. 60 = 0^2 * a + 0 * b + c -> c = 60
2. 5 = (150^2) * a + 150 * b + c -> 22500a + 150b + 60 = 5 -> 22500a + 150b = -55
Next, we can find the equation of the parabola by solving this system of equations. We can do this by using the second point we were given, (150, 5), along with a point on the axis of symmetry of the parabola.
Let's choose the coordinate (75, h) as our second point since it represents the midpoint between the center of the bridge and a supporting tower. We can find h by substituting the coordinate (75, h) into the equation of the parabola, yielding the following equation:
h = (75^2) * a + 75 * b + 60
Now we can substitute the known values from the point (150, 5) into this equation:
5 = (75^2) * a + 75 * b + 60
Simplifying this equation, we get:
(75^2) * a + 75 * b = -55
This equation can be rewritten as:
22500a + 75b = -55
We now have a system of equations:
22500a + 150b = -55
22500a + 75b = -55
To solve this system, we can subtract the second equation from the first:
22500a + 150b - (22500a + 75b) = -55 - (-55)
Simplifying, we have:
75b = 0
Therefore, b = 0.
Substituting b = 0 into the first equation, we get:
22500a + 150(0) = -55
22500a = -55
Dividing both sides by 22500, we find:
a = -55 / 22500
Simplifying this expression, we get:
a = -1 / 409
We now have the values of a, b, and c. The equation of the cable is given by:
y = (-1/409)x^2 + c
Substituting c = 60 into this equation, we have:
y = (-1/409)x^2 + 60
To find the lengths of the vertical supporting rods from the cable to the roadway, we need to determine the distances between the parabolic curve and the roadway at intervals of 50 meters from the center of the bridge to a supporting tower.
Let's calculate the distances at intervals of 50 meters:
- For x = 0 (center of the bridge):
- y = (-1/409)(0)^2 + 60 = 60
- Vertical supporting rod length = 60 - 5 (60 meters above the roadway) = 55 meters
- For x = 50:
- y = (-1/409)(50)^2 + 60 ≈ 49.88
- Vertical supporting rod length = 49.88 - 5 = 44.88 meters
- For x = 100:
- y = (-1/409)(100)^2 + 60 ≈ 34.12
- Vertical supporting rod length = 34.12 - 5 = 29.12 meters
- For x = 150 (supporting tower):
- y = (-1/409)(150)^2 + 60 ≈ 24.93
- Vertical supporting rod length = 24.93 - 5 = 19.93 meters
Note: The approach detailed above assumes that the parabolic shape of the cable can be modeled accurately using the given information.
To solve this problem, we need to find the equation of the parabola that represents the shape of the cable. Once we have the equation, we can determine the lengths of the vertical supporting rods from the cable to the roadway at the given intervals.
Let's consider the given information:
1. The supporting towers are 300 meters apart.
2. The highest point of the cable is 60 meters above the roadway.
3. The lowest point of the cable is 5 meters above the roadway.
To find the equation of the parabola, we need to use the standard form of a parabolic equation, which is y = a(x-h)^2 + k. In this equation, (h, k) represents the vertex of the parabola.
Let's assume the vertex of the parabola is at point (x,y) = (0, 5) since the lowest point of the cable is 5 meters above the roadway. So, k = 5.
Next, we need to find the value of "a" in the equation. Since the highest point of the cable is 60 meters above the roadway, we can substitute the point (150, 60) into the equation:
60 = a(150 - 0)^2 + 5
55 = 22500a
a = 55/22500
a = 11/4500
Now that we have the values of "a" and "k," we can write the equation of the parabola:
y = (11/4500)x^2 + 5
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 need to substitute different values of x into the equation and calculate the corresponding y-values.
The center of the bridge is the midpoint between the two supporting towers, which is 150 meters in this case. From this point, we can calculate the y-values at intervals of 50 meters in both directions.
Substituting x = 100 into the equation:
y = (11/4500)(100)^2 + 5
y ≈ 110.11
Substituting x = 150 into the equation:
y = (11/4500)(150)^2 + 5
y = 110
Substituting x = 200 into the equation:
y = (11/4500)(200)^2 + 5
y ≈ 110.89
Therefore, the lengths (vertical distance) of the supporting rods from the cable to the roadway at intervals of 50 meters from the center of the bridge to a supporting tower are approximately:
1. At x = 100 meters from the center: 110.11 - 60 = 50.11 meters
2. At x = 150 meters (center of the bridge): 110 - 60 = 50 meters
3. At x = 200 meters from the center: 110.89 - 60 = 50.89 meters
draw a diagram. If we put the vertex of the parabola at (0,5) then we have
y = ax^2 + 5
y(150) = 60
now you can find a, and then work on
y(0), y(50), ... y(150)