A new wind farm manufacturer is undertaking to optimise the size of its wind turbine generator iWTG}. At a blade diameter of 131m and a hub height of Bdm, the Altusl WTG is so far their prize model. Like all Altus WTG, it delivers a Betz limitlike power curve over all hub height wind speeds it intercepts. This means that when one writes the power output as a function of wind speed, one finds, for all of their turbines, that P: ov], wherev} is a function of v only. The constant, o is the same for all their WTG. The designers want to reconfigure the Altus to deliver maximal payback economics. The 131m model costs 3m EUR. Another version, the lm version {with hub height = Em] costs 1.94Bm EUR. The Altus 1 delivers BMW of power when its hub height wind speed reaches T mfs. The velocity profile, viz] fits the functional description given in your lectures: >v: = Vln{g:d] forzed and zd r: zo Where NE a reference speed, 20 is a characteristic roughness length, and dis an offset. Here they are given by 3 mfs, 11m, 10m respectively. We may also assume the longterm average speed at z is given by v2. We assume the WTG runs nonstop, round the clock at the output level, Piv] for the average speed at the hub height in question. The price of electricity {SfMWh} is also assumed to be constant. The engineers at Altus have agreed that to simplify their design, they can set all hub heights to half the rotor diameter plus some headroom, which they are holding constant at lm. They also know that the total cost of a WTG is proportional to the hub height (when rotor diameter is constant), while it is proportional to the square of the rotor diameter (when hub height held constant). Their goal is to find the hub height and rotor diameter that minimises the payback period. a) What is the value of a? b) Derive an expression, with all relevant constants, for the total cost, TC of the Altus WTG's as a function of hub height, h. c) Derive an expression for the function you are minimising in its simplest form. d) Find the hub height and diameter that minimizes the payback time for the Altus