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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

$$ Ely^n = mg(L - x) + \dfrac{1}{2}pg(L - x)^2 $$

$ E $ and $ l $ 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.

(a) $$

\begin{array}{l}{f(x)=\frac{m g}{6 E I}(L-x)^{3}+\frac{\rho g}{24 E I}(L-x)^{4}+\left(\frac{m g}{2 E I} L^{2}+\frac{\rho g}{6 E I} L^{3}\right) x} \\ {+\left(-\frac{m g}{6 E I} L^{3}-\frac{\rho g}{24 E I} L^{4}\right)}\end{array}

$$

(b) $$

f(L)=(8 m+3 \rho L) \frac{g L^{3}}{24 E I}

$$

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