The displacement hhh (in metres) of a specialized geological probe from a fixed baseline is modelled by the equation
h(x)=12ln(x+2)−14x2−4,x>−2 h(x) = 12\ln(x + 2) - \frac{1}{4}x^2 - 4, \quad x > -2 h(x)=12ln(x+2)−41x2−4,x>−2where x x\,x represents the horizontal distance in kilometres from a reference station. The curve crosses the baseline (h=0h=0h=0) at point P P\,P and point QQQ, as shown in the model, where P P\,P has a negative xxx-coordinate and Q Q\,Q has a positive xxx-coordinate.
Show that the xxx-coordinate of P P\,P lies in the interval [−0.6,−0.5][-0.6, -0.5][−0.6,−0.5].
The curve crosses the baseline at point Q Q\,Q for some x>0x > 0x>0. Using the iterative formula
xn+1=48ln(xn+2)−16with x1=10 x_{n+1} = \sqrt{48\ln(x_n + 2) - 16} \quad \text{with } x_1 = 10 xn+1=48ln(xn+2)−16with x1=10(i) find, to 4 decimal places, the value of x2x_2x2. (ii) find, by continued iteration, the xxx-coordinate of QQQ. Give your answer to 4 decimal places.
The curve has a maximum vertical displacement at point MMM.
Using calculus and showing each stage of your working, find the exact xxx-coordinate of MMM.