If a diver of mass $m$ stands at the end of a diving board with length $L$ and linear density $p$, then the board takes on the shape of a curve $y = f(x)$, where $$EIy''=mg(L-x)+\frac{1}{2}pg(L-x)^2,$$ Here, $E$ and $I$ are positive constants that depend on the material of the board, and $(g < 0)$ is the acceleration due to gravity.

(a) Find an expression for the shape of the curve (in other words, determine the equation for $y$.

$y =$

(b) Using $f(L)$, find the distance below the horizontal at the end of the board.

Distance below horizontal at the end =