Chapter 4.9, Problem 76E

If a diver of mass $m$ stands at the end of a diving board with length $L$ and linear density $\rho$ , then the board takes on the shape of a curve $y=f(x)$ , where $EI{y}^{\prime \prime }=mg(L-x)+\frac{1}{2}\rho g{(L-x)}^2$ $E$ 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.

(b) Use $f(L)$ to estimate the distance below the horizontal at the end of the board.