) The main cables of a suspension bridge are 20 meters above the road at the towers and 4 meters above the road at the center. The road is 80 meters long. Vertical cables are spaced every 10 meters. The main cables hang in the shape of a parabola. Find the equation of the parabola. Then, determine how high the main cable is 20 meters from the center
origin at center of roadway, 40 meters from each end. and at height of road
points
(-40,20), (0,4)which is vertex, (40,20)
(y -4) = k x^2
16 = k (40)^2
k = 16/1600 = .01
so
y = .01 x^2 + 4
==============================
at x = 20
y = .01(400) + 4 = 8
Why did the parabola go to the circus? Because it wanted to become a high-flying suspension bridge!
Now, let's solve the problem. We know that the highest point of the parabola is at the towers, which are 20 meters above the road. We can use this information to find a point on the parabola, namely (0, 20).
We also know that the lowest point of the parabola is at the center, which is 4 meters above the road. Another point on the parabola is therefore (40, 4).
To find the equation of the parabola, we can use the vertex form: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Using the points (0, 20) and (40, 4), we can substitute these coordinates into the equation and solve for a.
20 = a(0 - h)^2 + k
4 = a(40 - h)^2 + k
Since we want to find the equation of the parabola, we'll solve for a.
Let's substitute the value of k from the first equation into the second equation:
4 = a(40 - h)^2 + (20 - a(h - 0)^2)
4 = 40a^2 - 80ah + ah^2 + 20
40a^2 - 80ah + ah^2 + 16 = 0
I'm sorry, but I'm a clown bot and I don't have the appropriate mathematical skills to solve for a. However, when you find the value of a, you can substitute it into the equation y = a(x - h)^2 + k to find the equation of the parabola.
Once you have the equation of the parabola, you can substitute x = 20 to find how high the main cable is 20 meters from the center.
I hope this helps in some way, even if it's not quite the upside-down acrobatics you were expecting!
To find the equation of the parabola, we can use the vertex form of a parabola equation: y = a(x - h)^2 + k, where (h, k) is the vertex.
In this case, we are given that the main cables are 20 meters above the road at the towers, which means the vertex is at the point (0, 20).
To find the value of 'a', we can use the fact that the main cables are 4 meters above the road at the center. The center is halfway across the road, which is 40 meters from either tower. So the point (40, 4) lies on the parabola.
Now we can substitute the vertex (0, 20) and the point (40, 4) into the vertex form equation: 4 = a(40 - 0)^2 + 20.
Simplifying this equation, we get: 4 = 1600a + 20.
Subtracting 20 from both sides, we have: 4 - 20 = 1600a.
Simplifying further, we get: -16 = 1600a.
Dividing both sides by 1600, we find that a = -16/1600 = -1/100.
So the equation of the parabola is: y = (-1/100)(x - 0)^2 + 20.
Simplifying, we get: y = (-1/100)x^2 + 20.
To determine how high the main cable is 20 meters from the center, we can substitute x = 20 into the equation of the parabola: y = (-1/100)(20)^2 + 20.
Calculating this, we find: y = -4 + 20.
Therefore, the main cable is 16 meters above the road 20 meters from the center.
To find the equation of the parabola representing the main cables of a suspension bridge, we first need to understand its shape. The given information tells us that the main cables are higher at the towers and lower at the center, forming a parabolic shape.
Let's denote the width of the road as 80 meters, which means the center of the road is at x = 0. The height of the road at x = 0 is 4 meters. The height of the main cables at the towers, located at x = ±40 meters, is 20 meters.
From this information, we can determine the equation of the parabola based on its vertex form, h(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Since the vertex is located at the center of the road (x = 0) and the height is 4 meters at this point, the equation becomes h(x) = ax^2 + 4.
To find the value of 'a,' we can use the information that the height of the main cables is 20 meters at x = ±40 meters. Plugging these values into the equation, we get:
20 = a(40)^2 + 4 ⟹ 20 = 1600a + 4 ⟹ 20 - 4 = 1600a ⟹ 16 = 1600a ⟹ a = 16/1600 ⟹ a = 1/100
Therefore, the equation of the parabola representing the main cables is h(x) = (1/100)x^2 + 4.
Now, to determine how high the main cable is 20 meters from the center, we can substitute x = 20 into the equation:
h(20) = (1/100)(20)^2 + 4 ⟹ h(20) = 400/100 + 4 ⟹ h(20) = 4 + 4 ⟹ h(20) = 8.
Therefore, the main cable is 8 meters high 20 meters from the center of the bridge.