The vertical height, h h\,h metres, of a buoyant sensor bobbing in a turbulent drainage pipe relative to a fixed datum is modelled by the equation
h(t)=4sin2t−4t+10,t>0 h(t) = 4 \sin 2t - 4t + 10, \quad t > 0 h(t)=4sin2t−4t+10,t>0where t t\,t is the time in minutes and t t\,t is measured in radians.
The sensor first hits a submerged blockage at the point PPP, where h=0h = 0h=0. Given that the ttt-coordinate of P P\,P is α\alphaα:
Show that α \alpha\,α lies between 1.8 and 1.9.
The Newton-Raphson iteration formula
tn+1=4tncos2tn−2sin2tn−54cos2tn−2 t_{n+1} = \frac{4t_n \cos 2t_n - 2 \sin 2t_n - 5}{4 \cos 2t_n - 2} tn+1=4cos2tn−24tncos2tn−2sin2tn−5can be used to find an approximate value for α\alphaα.
Using this iteration formula with t1=1.7t_1 = 1.7t1=1.7, find, to 4 decimal places, the value of: (i) t2t_2t2 (ii) t5t_5t5
The points Q Q\,Q and R R\,R represent the times at which the sensor reaches its first and second local minimum heights respectively for t>0t > 0t>0.
Using calculus, find the exact value of the ttt-coordinate of Q Q\,Q and the exact value of the ttt-coordinate of RRR.