A sensor monitors the motion of a survey buoy in a harbor. For t≥1t \ge 1t≥1, where ttt is the time in minutes after a fixed reference point, the horizontal velocity of the water uuu (in knots) and the height of the buoy hhh (in metres) above the seabed are modelled by the parametric equations
u=16−(2t−6)2u = 16 - (2t - 6)^2u=16−(2t−6)2 h=12−5(2t−6)15h = 12 - 5(2t - 6)^{\frac{1}{5}}h=12−5(2t−6)51
Local maximum and minimum heights of the buoy occur when the horizontal velocity of the surrounding water is zero.
The sensor begins recording at 09:00, at which time t=1t = 1t=1 and the buoy is at a local maximum height.
(i) Use the model to find the height of the buoy at 09:00.
(ii) Find the time at which the first local minimum height occurs after 09:00.
(iii) Find the height of the buoy at this local minimum.
Use the model to find the height of the buoy when the horizontal velocity of the water is at its maximum.
Comment on the validity of the model.
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