$2.$ A variable$-$speed HAWT with rotor diameter of $80$ m operates with cut$-$in speed of $\backslash (4\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ cutout speed of $\backslash (30\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at wind speed probability distribution of Weibull $\backslash ((\backslash$mathrm$\{$c$\}=9,\backslash$mathrm$\{$k$\}=2\mathrm{.}0\backslash )).$ The maximum power coefficient for this turbine is $0\mathrm{.}5$ and mechanical efficiency is $0\mathrm{.}9,$ calculate the following:

a$)$ The appropriate generator rated power $($MW$)$ to generate exactly $9$ GWh per year. $(5$ points$)$

b$)$ Using the above generator, what is the power output at wind speeds of $\backslash (11\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\},15\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ and $\backslash (32\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )?(4$ points$)$

c$)$ How many hours per year does the turbine operate? $(3$ points$)$