$3.$ A variable$-$speed HAWT with rotor diameter of $110$ m operates with cut$-$in speed of $\backslash (4\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ cut$-$out speed of $\backslash (30\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and uses a generator with rated power of $5$ MW $.$ The maximum power coefficient for this turbine is $0\mathrm{.}5$ and mechanical efficiency is $0\mathrm{.}92.$ Calculate the following:

a$)$ Available wind power density $\backslash (\backslash$left$[\backslash$mathrm$\{$W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$right$]\backslash )$ at an onshore location with wind speed distribution of Weibull $(\backslash (\backslash$mathrm$\{$c$\}=9,\backslash$mathrm$\{$k$\}=2\mathrm{.}0)\backslash )$ and an offshore location with wind speed distribution of Weibull $(\backslash (\backslash$mathrm$\{$c$\}=14,\backslash$mathrm$\{$k$\}=2\mathrm{.}2\backslash )).(4$ points$)$

b$)$ Capacity factors obtained using this turbine during a year for the above onshore and offshore locations? $(6$ points$)$