Answer:
The height of the arch at its center is 250/9 or about 27.78 feet.
Step-by-step explanation:
We can write an equation to model the parabolic arch.
Let the left-most point of the arch be the origin (0, 0).
Since the bridge has a span of 100 feet, the right-most point must be (0, 100).
We can use the factored form of a quadratic:
![y=a(x-p)(x-q)](https://tex.z-dn.net/?f=y%3Da%28x-p%29%28x-q%29)
Where <em>p</em> and <em>q</em> are the <em>x-</em>intercepts.
Our <em>x-</em>intercepts are <em>x </em>= 0 and <em>x </em>= 100. Hence:
![y=ax(x-100)](https://tex.z-dn.net/?f=y%3Dax%28x-100%29)
At a point 40 feet from the center, the height of the arch is 10 feet.
The center is <em>x</em> = 50. So, a point 40 feet from the center can be either <em>x</em> = 10 or <em>x</em> = 90.
So, for instance, when <em>x</em> = 10, <em>y</em> = 10. Substitute and solve for <em>a: </em>
<em />
<em />
So:
![\displaystyle a=-\frac{1}{90}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20a%3D-%5Cfrac%7B1%7D%7B90%7D)
The same value will result if we let <em>x</em> = 90 and <em>y</em> = 10.
Hence, our equation is:
![\displaystyle y=-\frac{1}{90}x(x-100)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%3D-%5Cfrac%7B1%7D%7B90%7Dx%28x-100%29)
The height of the arch at its center will be when <em>x</em> = 50. Hence:
![y(50)=\displaystyle -\frac{1}{90}(50)((50)-100)=\frac{250}{9}\approx 27.78\text{ feet}](https://tex.z-dn.net/?f=y%2850%29%3D%5Cdisplaystyle%20-%5Cfrac%7B1%7D%7B90%7D%2850%29%28%2850%29-100%29%3D%5Cfrac%7B250%7D%7B9%7D%5Capprox%2027.78%5Ctext%7B%20feet%7D)