December 2007

### Finding the maximal efficiency

We want to calculate the maximum value of windmill efficiency $P/P_0$. Write $y$ for $P/P_0$ and $x$ for $V/U$. We get $$y = \frac{1}{2}(1-x^2)(1+x).$$ Differentiating we get $$dy/dx = \frac{1}{2}(1 - 2x - 3x^2).$$ The maximum occurs when $dy/dx = 0$, in other words when $$3x^2 + 2x - 1 = 0.$$ This happens for $x_0=1/3$ and $x_1=-1$. Discounting the negative solution we get a maximal efficiency of $$1/2(1-x_0^2)(1+x_0) = 16/27.$$

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