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 being $6\, m$. Then, the length of supporting wire attached to the roadway $18\, m$ from the middle is

• A. 10.02 m
• B. 9.11 m
• C. 10.76 m
• D. 12.06 m

Answer 1

Answer: (B)

Since, wires are vertical. Let equation of the parabola is in the form $x^{2}=4$ ay $ \ldots$ (i) Focus is at the middle of the cable and shortest and longest vertical supports are $6 \,m$ and $30 \,m$ and roadway is $100 \,m$ long.

Clearly, coordinate of $Q (50,24)$ will satisfy eq. (i)
$(50)^{2}=4 a \times 24 \Rightarrow 2500=96 a $
$\Rightarrow a =\frac{2500}{96}$
Hence, from eq. (i), $x^{2}=4 \times \frac{2500}{96} y $
$\Rightarrow x^{2}=\frac{2500}{24} y$
Let $P R=k \,m$
$\therefore $ Point $P (18, k )$ will satisfy the equation of parabola i.e.,
From eq. (i), $(18)^{2}=\frac{2500}{24} k$
$ \Rightarrow 324=\frac{2500}{24} k$
$\Rightarrow k =\frac{324 \times 24}{2500}=\frac{324 \times 6}{625}=\frac{1944}{625} $

$\Rightarrow k =3: 11$
$\therefore $ Required length $=6+ k =6+3.11=9.11 m$ (approx.)