Misc 3
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
Cable is always facing upwards with the axis vertical
Since, the axis is positive y-axis,
its equation is
x2 = 4ay
First, we find coordinates of point B
Given length of cable = 100 m
Hence, AB = 100 m
So, BC = 2 = 100 2 = 50 m
Also,
DB = 30 6 = 24 m
Hence point B is B (50, 24)
Now,
Since point B(50, 24) lies on the parabola
Putting x = 50, y = 24 in equation of parabola
x2 = 4ay
(50)2 = 4a (24)
2500 = 4a 24
4a 24 = 2500
4a = 2500 24
4a = 625 6
a =
Now, we need to length of a supporting wire attached to the roadway 18 m from the middle
Hence OR = 18 m,
We need to find QP
Let QP = d
So, QR = QP 6 = d 6
So, coordinates of point Q is Q(18, d 6)
So, coordinates of point Q is Q(18, d 6)
Since point Q lies on parabola
it will satisfy the equation of parabola
Equation of parabola is
x2 = 4ay
Putting x = 18 & y = d 6 & a = 625 24
182 = 4 625 24 (d 6)
18 18 = 4 625 24 (d 6)
18 18 24 4 625 = (d 6)
(d 6) = 18 18 24 4 625
(d 6) = 3.11
d = 6 + 3.11
d = 9.11
So, QP = 9.11 m
Length of support is 9.11 m.

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.