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 30m and the shortest wire being 6m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
Given that the cable hangs in the form of a parabola.
It is told that the length of the shortest wire supported is 6.
It is clear that the vertex of the parabola is S(0, 6).
We know that the equation of the parabola having (0, a) as vertex is x2 = 4b(y - a)
Let us assume the equation of the parabola is x2 = 4a(y - 6).
It is told that the road way is 100m long. We usually give maximum support at the middle of the road i.e. at 50m. At the 50m of the road way, the length of the support wire used is 30m.
It is clear that the point (50, 30) lies on parabola, substituting this point in the equation of the parabola, we get,
⇒ (50)2 = 4a(30 - 6)
⇒ 2500 = 4a(24)
⇒ 96a = 2500
⇒ 
.
The equation of the parabola is 
.
We need to find the length of the support that is needed to give at the 18m in the roadway.
Let us assume the length of the support is l m,
We have a point (18, l) on the parabola, substituting in the equation we get,
⇒ 
⇒ 
⇒ 3.11 = l - 6
⇒ l = 9.11m
∴The length of the support required is 9.11m.