$(30$ points$)$ A parabolic three$-$hinged arch $($points $A,B$ and $C$ are hinges$)$ has the dimensions and

loading configuration indicated in the figure below $(\{$:$l={200}^{\text{'}},h={32}^{\text{'}},q=7\frac{k}{f}t,P=480$kips$).$

Please note that the distributed load $q$ is facing down over the left half of the structure and up over

the right half. The concentrated force $P$ is acting at the crown $C.$ The expression for the parabolic

arch with respect to the coordinate system shown in the figure is: $y=\frac{4h{x}^2}{l^2}$ and the corresponding

expression for the slope of the arch is: $tan=\frac{8hx}{l^2}$

a$.$ Compute the horizontal and vertical components of the reaction at the supports A and $B$ and

check whether the resultant of these two components is tangential to the arch at the location

of the two supports.

b$.$ Determine the functional form of the bending moment from $x=-{100}^{\text{'}}$ up to $x=+{100}^{\text{'}}$ and

sketch the resulting bending moment diagram over this range.