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=50−7sint h_B = 50 - 7\sin t hB=50−7sintThe height hD h_D\,hD metres of the dampener plate above the seabed at time t t\,t seconds is given by
hD=15−4cost h_D = 15 - 4\cos t hD=15−4costShow 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.
