An offshore wave-energy converter consists of a floating buoy and a submerged dampener plate that both move vertically due to ocean swells. Over a specific operating cycle, the height hB h_B\,hB metres of the buoy above the seabed at time t t\,t seconds is given by hB=50−7sinth_B = 50 - 7\sin thB=50−7sint The height hD h_D\,hD metres of the dampener plate above the seabed at time t t\,t seconds is given by hD=15−4costh_D = 15 - 4\cos thD=15−4cost
Show that the initial vertical distance between the buoy and the dampener plate is 39 metres.
Show that the distance d d\,d metres between the buoy and the dampener plate at time t t\,t is given by d=35+Rcos(t+α)d = 35 + R\cos(t + \alpha)d=35+Rcos(t+α) where R R\,R and α \alpha\,α are positive constants to be found. Give α \alpha\,α in radians to two decimal places.
Hence, find the minimum distance between the buoy and the dampener plate. Give your answer to the nearest centimetre.
Practise Edexcel A Level Maths 7.4 Simplifying a cos x +- b sin x with exam-style questions for A Level Maths. 19 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.