# A bridge is built in the shape of a parabolic arch. The bridge has a span of 50 meters and a maximum height of 40 meters. How do you find the height of the arch 10 meters from the center?

May 16, 2016

$33.6$ meters.

#### Explanation:

Referring to the highest point as the origin O and the the altitude

from O as the x-axis, the equation of the parabola is ${y}^{2} = 4 a x$

From the data given, the ends of the bridge are at $\left(40 , \pm 25\right)$.

So, ${25}^{2} = \left(4 a\right) \left(40\right)$. This gives a = 125/32.

When, at 10 meters from the center, $y = \pm 10$. So, the height

there is $40 - x$, when $y = \pm 10$.

From the equation of the parabola, ${10}^{2} = \left(4\right) \left(\frac{125}{32}\right) x$. So, x =

6.4. And so, the required height $= 40 - \frac{32}{5} = 33.6$ meters.

The graph for the arch is in a befitting frame.

graph{(y-40+0.064x^2)(y)=0[-25 25 -40 40]}