Question

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 wire being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle. 

Answer 1

Let X'OX be the bridge and PAQ be the suspension cable.
The suspension cable forms a parabola with the vertex at (0, 6).
Let the equation of the parabola formed by the suspension cable be  \[\left( x - 0 \right)^2 = 4a\left( y - 6 \right)\] 

It passes through P (−50, 30) and Q (50, 30). 

∴ \[2500 = 4a\left( 30 - 6 \right)\] 

⇒ \[4a = \frac{2500}{24}\] 

Putting the value of 4a in equation (1): 

\[x^2 = \frac{2500}{24}\left( y - 6 \right)\] 

Let LM be the supporting wire attached at M, which is 18 m from the mid-point (O) of the bridge.
Let the coordinates of L be (18, l).
It lies on the parabola (2).  

∴ \[{18}^2 = \frac{2500}{24}\left( l - 6 \right)\] 

\[\Rightarrow l = 9 . 11 m\] 

Hence, the length of the supporting wire attached to the roadway 18 m from the middle is 9.11 m.