solve all parts answer key given $!!!$A wind turbine with $100$ m diameter rotor has a wind cut$-$in speed of $3($m$)/($s$)$ and a cut$-$out speed

of $10($m$)/($sec$).$ It is placed in a region where the wind speed has a uniform probability, f$=0\mathrm{.}1$ for

wind speed f$=0$v$>10($m$)/($s$)$P$\_($ave $)=(\backslash$int P$\_($max$)($v$)$f$($v$)$dv$)/(\backslash$int f$($v$)$dv$)$Pmax$($v$)$vf$($v$)10($m$)/($s$)0$ and f$=0$ for v$>10($m$)/($s$).$

a$.$ Define average power of moving wind as

P$\_($ave $)=(\backslash$int P$\_($max$)($v$)$f$($v$)$dv$)/(\backslash$int f$($v$)$dv$)$

where Pmax$($v$)$ is the maximum power available to the turbine from the moving wind as a

function of the wind velocity v and f$($v$)$ is the probability function. The integrals are

evaluated over possible wind speeds. What is Pave for this wind turbine?

$[$Answer: Close to $1$ MW$.]$

b$.$ What is the maximum power the turbine can produce in a $10($m$)/($s$)$ wind $($sometimes

referred to as the rated power$)?[$Answer: Nearly $2\mathrm{.}5$ MW$.]$

c$.$ What is the capacity factor for this turbine? $[$Answer: $0\mathrm{.}40.]$