In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 400 m apart, and the lowest point of the suspension cables is 100 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the lowest point of the cable.
Since the vertex is set at (0,0), you have a parabola
y = ax^2
where y(200) = 100
a*40000 = 100
a = 1/400
y = 1/400 x^2
thank you steve
There are 2 towers that are each 10m high. A rope that is 15m long is strung between the tops of the towers. At its lowest point the rope sags 2.5m above the ground (see schematic diagram). How far apart are the towers?
To find the equation of the parabolic part of the suspension cables, we can start by setting up a coordinate system with the origin at the lowest point of the cable. Let's call this point (0, 0).
Given that the towers are 400 m apart, we can place them at (-200, 0) and (200, 0) in our coordinate system. The lowest point of the suspension cables is 100 m below the top of the towers, so its y-coordinate is -100.
Now, let's consider a point on the parabolic part of the cables, with coordinates (x, y). Since the shape of the suspension cables is parabolic, it means that the shape can be described by a quadratic equation of the form y = ax^2 + bx + c.
Since the vertex of the parabola is at the lowest point, the vertex formula gives us x = -b/(2a). In this case, the x-coordinate of the vertex is 0 since the origin is at the lowest point. So, we have x = -b/(2a) = 0, which means b = 0.
Now, let's find the value of a. We have two points on the parabolic part of the cables: (-200, -100) and (200, -100). Substituting these coordinates into the equation y = ax^2 + bx + c gives us:
-100 = a(-200)^2 + 0(-200) + c
-100 = a(40000) + c
-100 = 40000a + c ------(1)
-100 = a(200)^2 + 0(200) + c
-100 = a(40000) + c
-100 = 40000a + c ------(2)
From equations (1) and (2), we can see that 40000a + c is equal to -100 in both cases.
Now, let's solve this system of equations to find the values of a and c:
40000a + c = -100 ------(1)
40000a + c = -100 ------(2)
Subtracting equation (2) from equation (1), we get:
0 = 0
This means that the system of equations is dependent, meaning any value of a and c will satisfy both equations.
Therefore, we can choose any values for a and c. Let's choose a = 0.001 and c = -99.999, for example. Then the equation of the parabolic part of the suspension cables is:
y = 0.001x^2 - 99.999
Keep in mind that these values are arbitrary, and there are infinitely many equations that could describe the parabolic part of the suspension cables.