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 lines 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

Since,wires are vertical.Let equation of the parabola is in the form

Focus is at the middle of the cable and shortest and longest vertical supports are 6 m and 30 m and roadway in 100 m long.

Clearly,coordinate of Q(50,24) will satisfy Eq.(i)

$(50{)}^2=4a\times 24\Rightarrow 2500=96a\Rightarrow a=\frac{2500}{96}Hence,fromeq.(i),x^2=4\times \frac{2500}{96}y\Rightarrow x^2=\frac{2500}{24}yLetPR=km\therefore PointP(18,k\left)will\text{satisfy the equation of parabola i.e.,}\text{From Eq.(i),}\right(18{)}^2=\frac{2500}{24}\times k\Rightarrow 324=\frac{2500}{24}k\Rightarrow k=\frac{324\times 24}{2500}=\frac{324\times 6}{625}=\frac{1944}{625}k-=3.11\therefore Requiredlength=6+k=6+3.11=9.11m\left(approx\right)$