Binomial Theorem

Question

# The cable of a uniformly loaded suspension bridge hang 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 a roadway 18 m from the middle.

Solution

Let AOB be the cable of a uniformly loaded suspension bridge as shown in the figure.
Let AL and BM be the longest wires of length 30 m each. Let OC be the shortest wire of length 6 m and LM be the road way as shown in the figure.

The longest supporting wire = AL = BM = 30 m
The shortest supporting cable = OC = 6 m
Length of the roadway  = LM = 100 m
∴                           LC = CM = 50 m
Let O be the vertex and OY be the axis of the parabola
∴    Equation of the parabola is                                  ...(i)
ON = CM = 50
BN = BM - MN = BM - OC = 30 - 6 = 24
∴   Co-ordinates of point B are (50, 24)
Since it lies on the parabola (i)
∴
∴   From (i), the equation of the parabola is:
or                                                                      ...(ii)
Let the length of the supporting wire PQ at a distance of 18 m be h.
∴                             OR = CQ = 18 m
PR = PQ - RQ = PQ - OC = h - 6
Co-ordinates of point P are (18, h - 6)
Since it lies on parabola (ii)
∴

1944 + 3750 = 625 h