$CO$#$3,4,9$ Consider the dam and the sheet pile represented below. The structure is built on top of a soil layer. The permeability of the soil is $k=2\mathrm{.}5*{10}^{-4}c\frac{m}{s},$ and the saturated unit weight is ${}_{\text{s}\text{a}\text{t}}=18k\frac{N}{m^3}.$ The flow net of this cross$-$section is presented in the figure. Assuming that the dam, the sheet pile, and the rock contact below the soil are impermeable, answer the following questions:

a$)$ Calculate the total flow rate of water through the permeable layer per unit depth $($qtotal$)$ in $c\frac{m^3}{s}.$

b$)$ Calculate and plot in a graph the uplift pressure distribution below the dam. Suggestion: consider $6$ points distributed as indicated in the figure $(P1$ at $x=0m,P2$ at $x=10m,P3$ at $x=19\mathrm{.}9m($right before the sheet pile$),P4x=20\mathrm{.}1m($right after the sheet pile$),P5$ at $x=30m,$ and $P6x=40m).($Assume: ${}_w=10k\frac{N}{m^3}).$

c$)$ Calculate the critical hydraulic gradient $($icrit$)$ for the scenario presented in the figure.

d$)$ Do you expect erosion $($e$.$g$.,$ "boiling" or "heave"$)$ at the downstream toe of the dam with the current water levels? Consider the element located at the downstream toe of your flow net.

e$)$ Assuming that the downstream water level is constant, calculate the maximum upstream water level if we want to avoid erosion at the downstream toe of the dam.