Question

the lions gate bridge in vancouver bc is supported by two towers and two cables that join them. the central span between the towers is 472 metres and the height of the tower from the road bed is 49 metres. suppose the curve of the cable is graphed on a grid, with the origin on the road at the centre of the bridge and the lowest point of the cable 5 metres above the road. at a horizontal distance of 150 metres from the centre of the bridge, how high is the cable above the road to the nearest metre?

Explanation:

Step1: Assume the cable is a parabola

Since the parabola is symmetric about the y - axis and passes through the points $(- 236,49)$ and $(236,49)$ (half of 472 is 236) and the vertex is at $(0,5)$, the general form of the parabola is $y = ax^{2}+5$. Substitute $x = 236$ and $y = 49$ into the equation:
$49=a\times(236)^{2}+5$
$49 - 5=a\times(236)^{2}$
$44=a\times55696$
$a=\frac{44}{55696}=\frac{11}{13924}$
So the equation of the parabola is $y=\frac{11}{13924}x^{2}+5$.

Step2: Find the height at $x = 150$

Substitute $x = 150$ into the equation $y=\frac{11}{13924}x^{2}+5$:
$y=\frac{11}{13924}\times(150)^{2}+5$
$y=\frac{11\times22500}{13924}+5$
$y=\frac{247500}{13924}+5$
$y\approx17.78+5$
$y\approx23$

Answer:

23 metres