Problem $3$

Consider the arch shown below.

a$)$ Find the optimal shape $y(x).$

b$)$ Suppose that the distributed load could only be over the first or over the second half of the arch $($illustrated below$).$ Find the optimal shape $y(x)$ that minimizes the moment in the arch $($for load cases $1$ and $2$ not occurring simultaneously$).$

c$)$ With the new shape found in b$),$ where is the maximum height located, and what is its value? $($show your steps$)$