A three$-$blade wind turbine with a $90$m rotor diameter operates with a tip speed ratio of $10.$

$($a$)$ What is the efficiency?

$($b$)$ What fraction of the Betz limit is achieved?

$($c$)$ What is the actual power output in kW for a wind speed of $11$ m$/$s$?$

$($d$)$ For the wind speed in part $($c$),$ what is the actual blade tip velocity?

$($e$)$ For the wind speed in part $($c$),$ how long does it take the rotor to make one rotation?

$($f$)$ Draw a wind farm composed of $25(5$x$5$ array$)$ of these units. If these units displace the same power of coal$-$fired power from the portfolio, what is the annual CO$2$ reduction $($kilogram mass basis$)$ provided by this farm, assuming that $40\%$ of the units operate $75\%$ of the year, and $80\%$ of the units operate $20\%$ of the year, and $100\%$ of the units operate during the remaining peak.