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Problem $02\mathrm{.}120-$ Wind turbine

Wind energy has been used since $4000BC$ to power sailboats, grind grain, pump water for farms, and generate electricity. In the United States alone, more than $6$ million small windmills, most of them under $5$ hp $,$ have been used since the $1850$s to pump water. Small windmills have been used to generate electricity since $1900,$ but the development of modern wind turbines occurred toward the end of the twentieth century in response to the energy crises in the early $1970$s$.$ Regions with an average wind speed of above $5\frac{m}{s}$ are potential sites for economical wind power generation. A typical wind turbine has a blade span $($or rotor$)$ diameter of about $100$ m and generates about $3$ MW of power. Recent wind turbines have power capacities above $10$ MW and rotor diameters of over $150$ m $.$

Consider a wind turbine with an $80-$m$-$diameter rotor that is rotating at $18\mathrm{.}00$ rpm $($revolutions per minute$)$ under steady winds at an average velocity of $34\mathrm{.}00k\frac{m}{h}.$ Assume that the wind turbine has an efficiency of $35$ percent $($i$.$e$.,$ it converts $35$ percent of the kinetic energy of the wind to electricity$).$ Take the density of air to be $1\mathrm{.}20k\frac{g}{m^3}.$

Problem $02\mathrm{.}120.$a $-$ Power produced by wind turbine

Determine the power produced, in kW$.($Round the final answer to one decimal place.$)$

The power produced is

kW$.$