Question

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100m$ long is supported by vertical wires attached to the cable, the longest wire being $30m$ and the shortest being 6 m. Find the length of a supporting wire attached to the roadway $18m$ from the middle.

Answer 1

It is given that the uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100m$ long is supported by vertical wires attached to the cable where longest wire is $30m$ and shortest wire is $6m$.

Take origin of the coordinate plane as the vertex of the parabola and its vertical axis along $y$ axis.

Here, \(AB=30~m, OC=6~m\) and $BC=50m$.

The equation of the parabola with vertex at origin and axis along y axis is represented as, $x^2=4ay$

So, coordinates of point $A$ are $(50,24)$.

Since $(50,24)$ is a point on the parabola, it satisfies the equation of the parabola.

$(50{)}^2=4\times a\times 24$

$\Rightarrow a=\frac{50\times 50}{4\times 24}$

$\Rightarrow a=\frac{625}{24}$

Substitute the value of $a$ in (1),

$x^2=4\times \frac{625}{24}\times y$

$\Rightarrow 6x^2=625y$

Since $x$ coordinate is $18$,

$\Rightarrow 6(18{)}^2=625y$

$\Rightarrow y=\frac{18\times 18\times 6}{625}=3.11$ (Approximately)

Now,

$DF=DE+EF=3.11m+6m=9.11m$

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