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
$x^2=4 a y$...(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), $(18)^2=\frac{2500}{24} \times k$
$\rightarrow 324=\frac{2500}{24} k$
$\rightarrow k=\frac{324 \times 24}{2500}=\frac{324 \times 6}{625}=\frac{1944}{625}$
$\rightarrow k=311$
$\therefore$ Required length $=6+k=6+3.11=9.11 m$ (approx.)