Express 7cosθ+5sinθ7\cos \theta + 5\sin \theta7cosθ+5sinθ in the form Rcos(θ−α)R\cos(\theta - \alpha)Rcos(θ−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 light intensity in a controlled growth chamber. The intensity, III units, is modeled by the equation
I=10012+7cos(0.25t)+5sin(0.25t),0≤t≤24 I = \frac{100}{12 + 7\cos(0.25t) + 5\sin(0.25t)}, \quad 0 \le t \le 24 I=12+7cos(0.25t)+5sin(0.25t)100,0≤t≤24where ttt is the number of hours since the start of the growth cycle.
Using the model, determine: the minimum light intensity measured in the chamber, giving your answer to 2 decimal places,
the value of ttt, to 2 decimal places, when the light intensity first reaches 6.5 units.
