Question

The tower of a bridge, hung in the form of a parabola, have their tops $30$ metres above the road-way are $200$ metres apart. If the cable is $5$ metres above the road way at the centre of the bridge, then the length of the vertical supporting cable $30$ metres from the centre is

• A. $100m$
• B. $30m$
• C. $\frac{9}{4}m$
• D. $\frac{29}{4}m$

Answer 1

Answer: (D)

$\frac{29}{4}m$

Let us assume that lowest point of cable as origin and equation of parabola will be $x^2=4ay$
Given, tower of bridge is $30$ meters above the road-way and lowest point is $5$ meters above the road. Then in equation y-coordinate of top of tower is $25$ meters and x-coordinate is $100$ meters,from equation we get ${100}^2=4a\left(25\right)⟹a=100$
Now,length of the vertical supporting cable 30 metres from the centre is given by $y+5$ where $y$ is given by ${30}^2=4\left(100\right)\left(y\right)⟹y=\frac{9}{4}$,
Then vertical length of cable is $5+\frac{9}{4}=\frac{29}{4}$.