Reaching the Betz limit via the Newton's method for one (1)-dimensional optimisation

The $Betz$ theory states that a horizontal axis wind turbine can only extract less than $\frac{16}{27}$ ($59.3\%$) of the kinetic energy of the wind. The value $\frac{16}{27}$ ($0.\overline{592}$) is called the $Betz$ limit. In $1919$, $Albert$ $Betz$ used a method that is analytic to derive the $Betz$ theory and henceforth finding the $Betz$ limit. In the theory, he derived momentum equations of an actuator disc ($AD$) in the stream of wind. In this research, a $Newton's$ method scheme is used to reach the $Betz$ limit. For the axial induction factor denoted by either $a$ or $a_{*}$ we used a reasonable initial point guess of $0.259$. The results agree well with the analytical results of the $Betz$ theory. From the $Newton's$ method scheme results corresponding to $a$ and $a_{*}$, the relative errors in comparison with the $Betz$ limit are $4\%$ and $0.9\%$, respectively. The small amount of errors shows the possibility of reaching the $Betz$ limit using the $Newton's$ method scheme. If alternate power coefficients [$C_{p}(a)$ or $C_{p_{*}}(a_{*})$] are found where the (exact) value ($a$ or $a_{*}$) cannot exactly be found analytically, the $Newton's$ method scheme would be an ideal way of estimating (or calculating) other proposed limits or the $Betz$ limit. This method yields extremely accurate results. Furthermore, for a quadratic equation, the quadratic formula would do. However, if the quadratic equation has floating-point numbers, the $Newton's$ method would be more appropriate.