Express 3cosx+2sinx3\cos x + 2\sin x3cosx+2sinx in the form Rcos(x−α)R\cos(x - \alpha)Rcos(x−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.
A researcher is monitoring the depth of water in a tidal harbor. The depth of the water, DDD meters, is modeled by the equation
D=154+3cos(0.4t)+2sin(0.4t),0≤t≤12 D = \frac{15}{4 + 3\cos(0.4t) + 2\sin(0.4t)}, \quad 0 \le t \le 12 D=4+3cos(0.4t)+2sin(0.4t)15,0≤t≤12where ttt is the number of hours after midnight.
Find, according to the model: the minimum depth of the water in the harbor, giving your answer to 2 decimal places,
the value of ttt, to 2 decimal places, when D=3.5D = 3.5D=3.5.
