Question 42 – 3.7.2 Numerical methods (A-level only) - Questions

The vertical height, $h\,$ metres, of a buoyant sensor bobbing in a turbulent drainage pipe relative to a fixed datum is modelled by the equation

\sin 2t - 4t + 10, \quad t > 0

where $t\,$ is the time in minutes and $t\,$ is measured in radians.

The sensor first hits a submerged blockage at the point $P$ , where $h = 0$ . Given that the $t$ -coordinate of $P\,$ is $α\alpha$ :

Show that $\alpha\,$ lies between 1.8 and 1.9.

The Newton-Raphson iteration formula

t_{n+1} = \frac{4t_n \cos 2t_n - 2 \sin 2t_n - 5}{4 \cos 2t_n - 2}

can be used to find an approximate value for $α\alpha$ .

Using this iteration formula with $t_1 = 1.7$ , find, to 4 decimal places, the value of: (i) $t_2$ (ii) $t_5$

The points $Q\,$ and $R\,$ represent the times at which the sensor reaches its first and second local minimum heights respectively for $t > 0$ .

Using calculus, find the exact value of the $t$ -coordinate of $Q\,$ and the exact value of the $t$ -coordinate of $R$ .