$6.$ Consider a wind turbine with $\backslash ($ n $\backslash )$ blades and diameter $\backslash ($ d $\backslash ).$ The power generated by the turbine depends on the air density, $\backslash (\backslash$rho $\backslash ),$ the wind velocity $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$w$\}\}\backslash ),$ the rate of rotation of the turbine axis, $\backslash (\backslash$omega $\backslash ),$ and the geometric parameters $\backslash ($ d $\backslash )$ and $\backslash ($ n $\backslash ).$

a$)$ Express the power dependency on the other parameters in dimensionless form. $(3$ points$)$

b$)$ If a model turbine of diameter $40$ cm develops $0\mathrm{.}5$ kW at sea level conditions when $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$w$\}\}=7\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and $\backslash (\backslash$omega$=1500\backslash$mathrm$\{$rpm$\}\backslash ),$ what is the power generated by a geometrically and dynamically similar prototype of diameter $15$ m in winds of $\backslash (18\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at an elevation of $1000$ m above sea level? $(1$ point$)$

c$)$ At what rate will the prototype rotate? $(1$ point$)$