one section of a suspension bridge has its weight uniformly distributed between twin towers that are 400 feet apart and rise 90 feet above the horizontal roadway. a cable strung between the tops of the towers has the shape of a parabola, and its center point is 10 feet above the roadway. 1) how do i find the equation of the parabola, and
2) how do I find the total length of the nine equally spaced cables?
make you sketch in such a way that the roadway is the x-axis and the center of the cable is along the y-axis
So the vertex of your parabola is (0,10) and your equation is
y = ax^2 + 10
we know (200,90) must be a point on it, so
90 = a(40000) + 10
a = 80/40000 = (1/500)x^2 + 10
2. length of the nine equally spaced cables?
where does that come from ? no mention of it in your problem
BTW, the cables of suspension bridges do NOT hang in the shape of a parabola. The hang in the shape of a "CATENARY".
1) Well, finding the equation of the parabola can be quite the bridge-toll task. But fear not, I'm here to guide you through it. Let's assume the equation of the parabola is in the form y = ax^2 + bx + c.
To start, we need to find the vertex of the parabola. The vertex is the lowest point, and in this case, it lies 10 feet above the roadway. So, the y-coordinate of the vertex is 10.
Since the towers are 400 feet apart, the x-coordinate of the vertex is at the midpoint, which is (400/2) = 200.
Now, let's plug in these coordinates into the equation:
10 = a(200^2) + b(200) + c
We also know that the vertex lies on the parabola, so the equation holds true:
10 = a(0^2) + b(0) + c
Simplifying, we arrive at two equations:
c = 10 and 200^2 a + 200b + c = 10
Next, we consider the height of the towers. Since the towers rise 90 feet above the roadway, when x = 0, y = 90. Substituting these values:
90 = a(0^2) + b(0) + c
Now we have three equations:
a(200^2) + 200b + 10 = 10
a(0^2) + b(0) + 10 = 90
Simplifying some more, we get:
a(200^2) + 200b = 0
10 = 90
Now, my dear friend, this implies that the left and right towers are not the same height and we have stumbled upon an unsolvable situation! But fear not, I jest. It seems I made a mistake, as we can't find a valid set of values for a, b, and c. So, let's retrace our steps and try again.
2) As for the total length of the nine equally spaced cables, it seems we need to utilize a little bit of arithmetic here. If we divide the span of 400 feet into nine equal parts, each part will have a length of (400/9) feet.
Now let's add up these nine equal parts:
(400/9) + (400/9) + (400/9) + (400/9) + (400/9) + (400/9) + (400/9) + (400/9) + (400/9)
Combining like terms (or like spans in this case):
(9 * (400/9)) = 400 feet
Ah, delightful! The total length of the nine equally spaced cables is 400 feet. So, you can now "suspend" your worries and keep your "bridge" of humor intact.
To find the equation of the parabola, we can use the general equation for a parabola:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Step 1: Find the vertex of the parabola.
The vertex is the center point of the cable, given as (0, 10) in this case.
Step 2: Find the value of 'a' in the equation.
We know that the two towers are 400 feet apart, and the cable is 10 feet above the roadway. The tower heights at the top of the parabola are 90 feet. This information helps us find the value of 'a'.
Using the vertex form equation for a parabola:
y = a(x - h)^2 + k
When x = 200 (halfway between the towers), y = 90 (height of the towers).
90 = a(200 - 0)^2 + 10
80 = a(200^2)
80 = 40000a
a = 80/40000
a = 0.002
Therefore, the equation of the parabola is:
y = 0.002x^2 + 10
Now, let's move on to the second question.
To find the total length of the nine equally spaced cables, we need to find the length of one cable and then multiply it by 9.
Step 1: Find the length of one cable.
To do this, we need to find the length of the curve formed by the parabola.
The formula to find the length of a curve is given by the integral:
L = ∫[a, b] √(1 + (dy/dx)^2) dx
In this case, a = -200 and b = 200 (since the towers are 400 feet apart).
To find dy/dx (the derivative of y with respect to x), we differentiate the equation y = 0.002x^2 + 10:
dy/dx = 0.004x
Now, we can substitute this into the formula for length:
L = ∫[-200, 200] √(1 + (0.004x)^2) dx
This integral can then be evaluated to find the length of one cable.
Once you have this length, you can multiply it by 9 to find the total length of the nine equally spaced cables.
To find the equation of the parabola representing the cable, we can make use of the given information. Let's denote the vertex of the parabola as (h, k).
1) Finding the equation of the parabola:
The vertex form equation of a parabola is given by: y = a(x-h)^2 + k, where (h, k) represents the vertex.
We're given that the vertex is 10 feet above the roadway, so h = 0 and k = 10.
Now, let's find the value of a:
To do that, we need to consider two other points on the parabola that are equidistant from the vertex. Let's choose the points (x1, y1) and (x2, y2).
Point 1: Since the twin towers are 400 feet apart and rise 90 feet above the roadway, we can select (-200, 90) as one point because it is equidistant from the vertex as (200, 90).
Point 2: The other point that is equidistant from the vertex is at (200, 90).
Now we have three points:
(0, 10), (-200, 90), and (200, 90).
Substituting these points into the parabola equation, we can solve for a.
For the vertex point (0, 10):
10 = a(0-0)^2 + 10
10 = 10
For point 1 (-200, 90):
90 = a(-200 - 0)^2 + 10
90 = 40000a + 10
40000a = 80
a = 0.002
Now we can substitute the value of a into the equation to get the final equation of the parabola:
y = 0.002x^2 + 10
2) Finding the total length of the nine equally spaced cables:
To find the total length of the nine equally spaced cables, we need to find the lengths of all nine cables and then add them together.
Since the cable has the shape of a parabola, the cables between any two evenly spaced points on the parabola will all have the same length. So, we need to find the length of one cable and then multiply it by nine.
To find the length of one cable, we can use calculus integration.
Let's consider two neighboring points on the parabola, (x1, y1) and (x2, y2), where y1 and y2 represent the heights of the cable at those respective x-values.
The formula to find the length of a small segment of the cable is given by ds = sqrt(1 + (dy/dx)^2) * dx.
Here, (dy/dx) represents the derivative of y with respect to x.
Taking the derivative of the parabolic equation, we have dy/dx = 0.004x.
Now we can substitute dy/dx into the length formula.
ds = sqrt(1 + (0.004x)^2) * dx
ds = sqrt(1 + 0.000016x^2) * dx
Now, to find the total length of one cable between two neighboring points, we need to integrate the above expression between the two corresponding x-values.
The limits of integration will depend on the span of the bridge and the distance between the chosen points.
Since the twin towers are 400 feet apart, the distance between the chosen points will be 400/8 = 50 feet (as there are 9 intervals between 8 chosen points).
Integrating the expression with respect to x between -25 and 25 (as each cable length will range from -25 to +25):
Total length of one cable = ∫(-25 to 25) sqrt(1 + 0.000016x^2) dx
Once we have the length of one cable, we can multiply it by nine to get the total length of the nine equally spaced cables.