Question

1. suspension bridge a suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. the cables are parabolic in shape and are suspended from the tops of the towers. the cables touch the road surface at the center of the bridge. find the height of the cables at a point 100 meters from the center. (assume that the road is level.)

Explanation:

Step1: Set up the parabolic equation

Place the origin at the center of the bridge on the road - surface. The general form of a parabola opening upwards is $y = ax^{2}+b$. Since the parabola passes through the origin $(0,0)$, substituting $x = 0$ and $y = 0$ into the equation gives $0=a(0)^{2}+b$, so $b = 0$. The tops of the towers are at $x=\pm200$ (since the towers are 400 meters apart) and $y = 75$. Substituting $x = 200$ and $y = 75$ into $y=ax^{2}$ gives $75=a(200)^{2}$.

Step2: Solve for the coefficient $a$

From $75=a(200)^{2}$, we can solve for $a$ as $a=\frac{75}{200^{2}}=\frac{75}{40000}=\frac{3}{1600}$. So the equation of the parabola is $y=\frac{3}{1600}x^{2}$.

Step3: Find the height at $x = 100$

Substitute $x = 100$ into the equation $y=\frac{3}{1600}x^{2}$. Then $y=\frac{3}{1600}\times(100)^{2}=\frac{3\times10000}{1600}=\frac{300}{16}= 18.75$ meters.

Answer:

The height of the cables at a point 100 meters from the center is 18.75 meters.