Let's draw the situation presented in the problem:
In order to determine the radius of the circle that contains the bridge, we'll use the following formula:
[tex]r=\frac{h}{2}+\frac{w^2}{8h}[/tex]where h is the height of the arc and w is its width.
This formula is derived from the intersecting chords theorem:
[tex]a\cdot a=b\cdot c[/tex]Since in our case a is half of the width of the arc and b its height:
[tex]\frac{w}{2}\cdot\frac{w}{2}=h\cdot c[/tex][tex]\frac{w^2}{4}=h\mathrm{}c[/tex]dividing both sides by h:
[tex]\frac{w^2}{4h}=c[/tex]since the diameter of the circle is b+c, or in this case h+c:
[tex]d=h+\frac{w^2}{4h}[/tex]since the radius is half the diameter:
[tex]r=\frac{h}{2}+\frac{w^2}{8h}[/tex]Now, let's plug the data we were given into this formula:
[tex]r=\frac{3.3}{2}+\frac{23^2}{8\cdot3.3}[/tex][tex]r=1.65+\frac{529}{26.4}=1.65+0.0378=21.6878[/tex]So the radius of the circle will be 21.6878 ft.
