Problem $2$: $(50$pts$)$: A $15-$ft$-$high gravity dam is depicted in the figure. Assume the coefficient of friction between the $12ft-$ long dam base and the foundation is $0\mathrm{.}6,$ the specific gravity of concrete is $2\mathrm{.}65,$ and full uplift forces exist on the base of the dam. please:

a$)$ Draw free body diagram for the dam $($ please make sure the force locations are demonstrated correctly $({10}^{\text{'}})$

b$)$ If a force ratio against sliding of $1\mathrm{.}3$ is required. Determine the depth of water $(H)$ that can't be exceeded $($neglect earthquake and sedimentation forces$)(20\text{'}).$

c$)$ When the water behind the dam rises to a depth of $15ft,$ determine the resulant vertical forces per unit width and the distance from the center line of the resulant vertical force $({10}^{\text{'}})$

d$)$ Determine the foundation bearing pressure at the heel and toe $(($:${10}^{\text{'}}\}.$