The cables of the middle part of a suspension bridge are in the form of a parabola, and the towers supporting the cables are 600 ft apart and 100 ft high. If the cable, at its lowest is also at its midpoint.
see the post below this one, where a question is actually asked.
To find the equation of the parabolic cables, we can use the vertex form of a parabola:
y = a(x-h)^2 + k
Where (h, k) represents the vertex of the parabola and a is a constant that determines the shape of the parabola.
We're given that the towers supporting the cables are 600 ft apart, so the x-coordinate of the vertex is the midpoint between the two towers, which is (600/2) = 300 ft. Since the vertex is also at the lowest point of the cable, the y-coordinate of the vertex is the lowest point of the cable, which is the same as the y-coordinate of the midpoint.
Next, we need to determine the value of a to complete the equation. Since both ends of the cable are at the same height as the towers (100 ft), the cable passes through those two points, which are (0, 100) and (600, 100).
Let's calculate the value of a using these two points:
Using the vertex form, we substitute the coordinates (0, 100) into the equation:
100 = a(0 - 300)^2 + k
Simplifying, we know that (0 - 300)^2 is 90000 since -300 squared is 90000:
100 = a(90000) + k
Now, we substitute the coordinates (600, 100) into the equation:
100 = a(600 - 300)^2 + k
Simplifying, (600 - 300)^2 is 90000:
100 = a(90000) + k
Since both equations have k on the right side, the k values must be the same. So we now have:
100 = a(90000) + k
100 = a(90000) + 100
Subtracting 100 from both sides:
0 = a(90000)
a = 0
Now, substituting the value of a back into one of the earlier equations:
100 = 0(90000) + k
100 = k
So the value of k is 100.
Therefore, the equation of the parabolic cables is:
y = a(x-h)^2 + k
y = 0(x-300)^2 + 100
y = 100
The equation of the cables is simply y = 100, indicating that the cables are always at a height of 100 ft above the ground.