Chapter 3.9, Problem 64E

If a diver of mass m stands at the end of a diving board with length L and linear density $\text{ρ}$, then the board takes on the shape of a curve $y=f\left(x\right)$, where

$EIy\prime \prime =mg\left(L-x\right)+\frac{1}{2}\text{ρg}{\left(L-x\right)}^2$

E and I are positive constants that depend on the material of the board and $g\left(<0\right)$ 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.