$\backslash \{30$ pts$\backslash \}$ Background: The San Francisco Public Utility Company $($SFPUC$)$ Peninsula Watershed contains the highest concentration of rare, threatened, and endangered species in the nine$-$county Bay Area and is a State Fish and Game Refuge. Since the watershed is an environmentally sensitive area, it has been state and federally mandated that Pilarcitos Creek maintain a minimum water depth of $3$ meters.

Problem: Given the federal and state requirements for the minimum water depth, engineers for SFPUC were tasked with designing a concrete dam $($Figure $1)$ on Pilarcitos Creek. During the initial design phase, the engineers were concerned that the dam structure would slip along the solid foundation. Therefore, the engineers needed to determine the maximum allowable water depth $\backslash ((\backslash$mathbf$\{$H$\})\backslash )$ that the dam structure could resist before slipping on the foundation. The coefficient of static friction $\backslash (\backslash$left$(\backslash$mu$\_\{\backslash$mathrm$\{$s$\}\}\backslash$right$)\backslash )$ between the dam and the foundation is $0\mathrm{.}123.$ Assume no water seepage at the base of the dam and base your analysis on a unit length of dam $($in other words the width of the dam is $1$ meter into the paper$).$

Figure $1.$ Initial design of the concrete dam on Pilarcitos Creek.