Chapter 3.4, Problem 42ES

A new design for a wind turbine adjusts the length of the turbine blade to keep the generated power constant even if the wind speed changes by a small amount. Assume that the power $P$ (in watts) generated by the turbine can be expressed as $P=0.87{\mathcal{l}}^2{\nu }^3$ , where $\mathcal{l}$ is the length of the blade in meters and $\nu$ is the speed of the wind in meters per second. Suppose the wind speed is increasing at a constant rate of $0.01{\text{ m/s}}^{\text{2}}$ and that the length of the blade adjusts to keep the generated power constant. Determine how quickly $\mathcal{l}$ is changing at the moment when $\mathcal{l}=16$ and $\nu =4$ .