A high-precision docking system in an aerospace facility uses two automated parallel assemblies, A A\,A and BBB, which oscillate vertically to dampen vibrations during a calibration cycle of 6π 6\pi\,6π minutes. At time t t\,t minutes after the cycle begins, the height hA h_A\,hA metres of assembly A A\,A above the hangar floor is given by
hA=26.8−7sint h_A = 26.8 - 7\sin t hA=26.8−7sintAt time t t\,t minutes after the cycle begins, the height hB h_B\,hB metres of assembly B B\,B above the hangar floor is given by
hB=7.4−5cost h_B = 7.4 - 5\cos t hB=7.4−5costShow that the initial vertical separation between assembly A A\,A and assembly B B\,B is 24.4 metres.
Show that the distance Δ \Delta\,Δ metres between the two assemblies at time t t\,t is given by
Δ=19.4+Rcos(t+α) \Delta = 19.4 + R\cos(t + \alpha) Δ=19.4+Rcos(t+α)where R R\,R and α \alpha\,α are positive constants to be found. Give R R\,R in surd form and α \alpha\,α in radians to two decimal places.
Hence, determine the minimum vertical distance between the two assemblies. Give your answer to the nearest centimetre.
