a$)$ The maximum water level is $120$ meters above the base, with a bottom width of $30$ meters and a width of $150$ meters at the maximum water level in a trapezoidal valley. Calculate the arch thickness of a concrete arch dam located $50$ meters below the surface and determine the arch reactions on the slopes for a unit body height.

$($The central angle of the arch will be taken as $2\mathrm{.}33$ radians, $\_$concrete$=420$ t$/$m$^2,$ and $\_$water$=1$ t$/$m$^3.)$

b$)$ Calculate the total volume of the concrete arch dam body.

c$)$ The tailwater of a hydroelectric power plant is located $180$ m below the maximum water level. The average flow rate is $320$x$10$ m$^3/$year$,$ and the total plant efficiency is $0\mathrm{.}85.$ Given that the friction factor in the penstock is $0\mathrm{.}02,$ the total local losses are $2\mathrm{.}5,$ the penstock length is $600$ m$,$ and the maximum flow rate can be $2\mathrm{.}0$ times the average flow rate, calculate the penstock diameter, wall thickness, average power, maximum power, and load factor.

$($k$=2,$ $\_$max$=1300$ kgf$/$cm$^2,$ E$=2\mathrm{.}1$x$10$ kgf$/$cm$^2)$