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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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##### Top Calculus 2 / BC Educators ##### Catherine R.

Missouri State University   ##### Kristen K.

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##### Top Calculus 2 / BC Educators ##### Catherine R.

Missouri State University   ##### Kristen K.

University of Michigan - Ann Arbor

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