The displacement of a buoy from its equilibrium position, in decimetres, is modeled by the function
h(t)=12cost−5sint h(t) = 12\cos t - 5\sin t h(t)=12cost−5sintwhere ttt is the time in minutes since the start of the observation.
Express h(t)h(t)h(t) in the form Rcos(t+α)R\cos(t + \alpha)Rcos(t+α), where R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of RRR and the value of α\alphaα, in radians, to 3 decimal places.
Let the stability index of the buoy be defined by S(t)=10−3h(4t)S(t) = 10 - 3h(4t)S(t)=10−3h(4t).
Using the answer to part (a), (i) write down the exact maximum value of S(t)S(t)S(t). (ii) find the smallest positive value of ttt for which this maximum value occurs, giving your answer to 2 decimal places.
