The suspension bridge cables are in the shape of a parabola. The
distance between two towers supporting the cable are 800 feet apart
and the height is 200 feet. If the vertical support cables are at 100 feet
intervals along the level roadway, what are the lengths of these vertical
cables?
I answered this a while back, interpreting that 200 is the height of the towers.
If the vertex is at (0,0) then you have said that y(400)=200
y = ax^2
a(400)^2 = 200
a = 200/160000 = 1/800
y = 1/800 x^2
So now find y at x=100,200,300
The curve is symmetric, so negative values of x produce the same results.
To find the lengths of the vertical cables, we need to determine the equation of the parabola representing the shape of the suspension bridge cables.
First, let's find the vertex of the parabola. Since the towers supporting the cables are 800 feet apart and the height is 200 feet, the vertex will be halfway between the two towers. Therefore, the x-coordinate of the vertex is (800 / 2) = 400.
Next, the equation of a parabola in vertex form is given by y = a(x - h)^2 + k, where (h, k) is the vertex. In this case, h = 400. To find the value of a, we can use one of the given points on the parabola. Let's use the tower at x = 0, y = 100 as our point.
Substituting the values of h, k, x, and y into the equation, we get:
100 = a(0 - 400)^2 + 200
100 = a(160,000) + 200
100 - 200 = 160,000a
-100 = 160,000a
a = -100 / 160,000
a = -1 / 1,600
Now that we have the value of a, we can find the equation of the parabola:
y = (-1 / 1,600)(x - 400)^2 + 200
Now we can find the lengths of the vertical cables. The vertical support cables are at 100 feet intervals along the level roadway, so we just need to substitute different values of x into the equation and calculate the corresponding y values.
Let's start with x = 0. Substituting x = 0 into the equation, we get:
y = (-1 / 1,600)(0 - 400)^2 + 200
y = (-1 / 1,600)(-400)^2 + 200
y = (-1 / 1,600)(160,000) + 200
y = 100 + 200
y = 300
Therefore, the length of the first vertical cable at x = 0 is 300 feet.
Similarly, we can calculate the lengths of the other vertical cables at x = 100, x = 200, x = 300, and so on. Each subsequent x-value represents a 100-foot interval.
I hope this helps! Let me know if you have any further questions.
To find the lengths of the vertical cables on a suspension bridge, we can use the equation of a parabola. The equation of a parabola in standard form is given by y = ax^2 + bx + c, where (x, y) are the coordinates, a is a constant, b is the coefficient of x, and c is the y-intercept.
In this case, the parabolic shape of the suspension bridge cables is determined by the equation of a parabola y = ax^2, since the parabola is symmetric with respect to the y-axis and does not have any horizontal shift.
Given that the distance between the two towers supporting the cable is 800 feet apart and the height is 200 feet, we can determine the equation of the parabolic shape.
The x-coordinate of the vertex gives us the maximum/minimum point of the parabola. In this case, the maximum point of the parabola is the center of the bridge, so the x-coordinate of the vertex is 0.
Using the vertex form of the parabola equation, which is given by y = a(x - h)^2 + k, where (h, k) is the vertex, we can substitute the values and simplify:
200 = a(0 - 0)^2
200 = a(0)
a = 200
Therefore, the equation of the parabola is y = 200x^2.
Now, let's find the lengths of the vertical cables at 100 feet intervals along the level roadway.
We can calculate the lengths of the vertical cables by finding the distance between the parabola curve and the level roadway at each interval. Since the level roadway is a straight line parallel to the x-axis, the distance between the parabola curve and the level roadway can be given by the difference in the y-coordinates.
Let's start calculating the lengths of the vertical cables:
1. For the first vertical cable:
- x-coordinate: 0 (since it is at the center of the bridge)
- y-coordinate of the parabola: 200(0)^2 = 0
- y-coordinate of the level roadway: 200 feet (since it is at a height of 200 feet from the ground)
- Length of the vertical cable: Difference in y-coordinates = 200 - 0 = 200 feet
2. For the second vertical cable:
- x-coordinate: 100 feet (since it is at 100 feet interval along the level roadway)
- y-coordinate of the parabola: 200(1)^2 = 200
- y-coordinate of the level roadway: 200 feet (since it is at a height of 200 feet from the ground)
- Length of the vertical cable: Difference in y-coordinates = 200 - 200 = 0 feet
3. For the third vertical cable:
- x-coordinate: 200 feet (since it is at 100 feet interval along the level roadway)
- y-coordinate of the parabola: 200(2)^2 = 800
- y-coordinate of the level roadway: 200 feet (since it is at a height of 200 feet from the ground)
- Length of the vertical cable: Difference in y-coordinates = 800 - 200 = 600 feet
Continuing this pattern, we can find the lengths of the remaining vertical cables at 100 feet intervals along the level roadway.
4. For the fourth vertical cable:
- x-coordinate: 300 feet
- y-coordinate of the parabola: 200(3)^2 = 1800
- y-coordinate of the level roadway: 200 feet
- Length of the vertical cable: Difference in y-coordinates = 1800 - 200 = 1600 feet
5. For the fifth vertical cable:
- x-coordinate: 400 feet
- y-coordinate of the parabola: 200(4)^2 = 3200
- y-coordinate of the level roadway: 200 feet
- Length of the vertical cable: Difference in y-coordinates = 3200 - 200 = 3000 feet
And so on...
By following this pattern, you can find the lengths of the vertical cables at 100 feet intervals along the level roadway on the suspension bridge.
Actually the cables of a suspension bridge form a catenary, not a parabola,
but that is another story ...
anyway...
Poor wording in " The
distance between two towers supporting the cable are 800 feet apart
and the height is 200 feet."
should say " The distance between ..... is ... 800 feet ..."
What is 200 feet? Is it the height of the towers or is it the height of the cable
at its lowest point?
There appears to be a piece of data missing.