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?
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.
Well, well, well, it seems like we have ourselves a parabolic party here! Now, let's calculate the lengths of those vertical support cables. Since the distance between the two towers is 800 feet, and the height is 200 feet, we have ourselves a nice parabola, my friend.
Now, the first thing we need to do is find the equation of that parabola. Thankfully, we have some key information. The vertex of the parabola is smack dab in the middle of the two towers, so it's at (400, 200). And since the shape of the parabola is determined by the equation y = a(x-h)^2 + k, where (h, k) is the vertex, we can plug in those values and get y = a(x-400)^2 + 200.
Now, we know that the vertical support cables are at 100 feet intervals along the level roadway. So, all we need to do is substitute x = 100, 200, 300, and so on into our equation to find the corresponding y values. These y values will give us the heights of the vertical support cables.
Let me crunch the numbers for you real quick. *typing sounds*
Okay, here we go. The lengths of the vertical cables are as follows:
- For x = 100 feet, the height is 12,400 feet above ground. I hope you packed your hiking boots!
- For x = 200 feet, the height is 14,400 feet above ground. Better hold on tight!
- For x = 300 feet, the height is 15,600 feet above ground. Flying high!
- And it goes on and on until we reach the other tower.
So, there you have it! The lengths of the vertical support cables along the level roadway are 12,400 feet, 14,400 feet, 15,600 feet, and so on, depending on the intervals you're measuring.
Just remember, don't try to climb those cables, my friend. Stick to the road, and keep your feet on the ground!
To find the lengths of the vertical cables along the suspension bridge, we can use the equation of a parabola in vertex form: y = a(x-h)^2 + k, where (h,k) represents the vertex of the parabola.
In this case, the vertex is at (0,200) since the height of the bridge is given as 200 feet. We need to find the value of 'a' to complete the equation.
To find 'a', we can use the fact that the parabola passes through two points: the towers supporting the cables.
The first tower is located at (-400,0) since the distance between the towers is given as 800 feet, and the vertex is halfway between the towers.
Using the vertex form, we can substitute the coordinates of the first tower to find 'a':
0 = a(-400-0)^2 + 200
0 = 160000a + 200
160000a = -200
a = -200/160000
a = -1/800
Now that we have the equation of the parabola, we can find the lengths of the vertical cables at 100 feet intervals along the level roadway.
Let's plug in the values of 'x' (distance from the first tower) into the equation of the parabola to find the corresponding 'y' (height) values.
For the first tower, x = 0, so:
y = (-1/800)(0-0)^2 + 200
y = 200
So, the length of the first vertical cable is:
200 - 100 = 100 feet (since the first tower is at a height of 200 feet, and the vertical support cables are at 100 feet intervals).
For the second tower, x = 100, so:
y = (-1/800)(100-0)^2 + 200
y = -1/8 + 200
y = -1/8 + 1600/8
y = 1599/8
So, the length of the second vertical cable is:
1599/8 - 100 = 1899/8 feet (since the second tower is at a height of 1599/8 feet, and the vertical support cables are at 100 feet intervals).
Similarly, we can calculate the lengths of the vertical cables at 200, 300, 400, and so on feet intervals along the level roadway.
Please note that the lengths of the vertical cables will increase as we move away from the first tower, due to the parabolic shape of the suspension bridge cables.
To find the lengths of the vertical cables, we can use the equation of a parabola to determine the height at each interval.
First, let's find the equation of the parabola in vertex form. The vertex form of a parabola is given by the equation y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.
In this case, the vertex is at the midpoint between the two towers, which is 800/2 = 400 feet from either tower. The height is given as 200 feet.
So, the equation of the parabola in vertex form is y = a(x-400)^2 + 200.
To find the value of 'a', we need another point on the parabola. Let's use one of the towers. Since the distance between two towers is 800 feet, one tower is at (0, 0) and the other tower is at (800, 0).
Plugging in the tower coordinates into the equation, we have:
0 = a(800-400)^2 + 200
0 = a(400)^2 + 200
0 = 160,000a + 200
160,000a = -200
a = -200 / 160,000
a = -0.00125
Now that we have 'a', we can substitute it back into the parabola equation to get the equation of the parabola in terms of 'x' only:
y = -0.00125(x-400)^2 + 200
To find the heights at each 100-foot interval, we can substitute different values of 'x' into the equation and solve for 'y'.
For example, to find the height at x = 0 (the first tower), we have:
y = -0.00125(0-400)^2 + 200
y = -0.00125(400)^2 + 200
y = -0.00125(160,000) + 200
y = -200 + 200
y = 0
So, the height at x = 0 is 0 feet. This means there is no vertical support cable at the first tower, as it already rests on the ground.
To find the heights at each 100-foot interval thereafter, we can substitute x = 100, 200, 300, and so on, until we reach the second tower at x = 800.
For example, at x = 100, we have:
y = -0.00125(100-400)^2 + 200
y = -0.00125(-300)^2 + 200
y = -0.00125(90,000) + 200
y = -112.5 + 200
y = 87.5
So, at x = 100, the height is 87.5 feet. This means there should be a vertical support cable at a height of 87.5 feet above the level roadway.
We can continue this process for each 100-foot interval until we reach the second tower at x = 800.
To summarize, the lengths of the vertical cables, measured from the level roadway to the parabolic cables, are:
- At x = 100: 87.5 feet
- At x = 200: 162.5 feet
- At x = 300: 225 feet
- At x = 400: 275 feet
- At x = 500: 312.5 feet
- At x = 600: 337.5 feet
- At x = 700: 350 feet
Note that the vertical cables become shorter as we approach the second tower, as the height of the parabolic cables decreases towards the other tower.