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

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

Answer 1

Answer: (B)

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

Clearly, coordinate of $Q(50,24)$ will satisfy Eq. (i)
$(50{)}^2=4a\times 24$
$\to 2500=96a\to a=\frac{2500}{96}$
Hence, from Eq. (i), $x^2=4\times \frac{2500}{96}y$
$\to x^2=\frac{2500}{24}y$
Let $PR=km$
$\therefore$ Point $P(18,k)$ will satisfy the equation of parabola i.e.,
From Eq. (i), $(18{)}^2=\frac{2500}{24}\times k$
$\to 324=\frac{2500}{24}k$
$\to k=\frac{324\times 24}{2500}=\frac{324\times 6}{625}=\frac{1944}{625}$
$\to k=311$
$\therefore$ Required length $=6+k=6+3.11=9.11m$ (approx.)