The profile of a suspended track for a high-speed transport system is modelled by the curve CCC. The path of the track starts at point P P\,P and ends at point QQQ, as defined by the parametric equations
x=12t2+2t+10 x = \frac{1}{2}t^2 + \frac{2}{t} + 10 x=21t2+t2+10 h=3t+12t h = 3t + \frac{12}{t} h=3t+t12where 1≤t≤61 \le t \le 61≤t≤6.
The horizontal distance from a sensor at the origin is x x\,x metres, and h h\,h is the height of the track above the ground in metres.
P P\,P is the point on the track where t=1t = 1t=1 and Q Q\,Q is the point where t=6t = 6t=6.
Safety regulations require that the difference in height between the start point P P\,P and the end point Q Q\,Q must be less than 6 metres. Show that the track meets this requirement.
Find an expression for dhdx\displaystyle \frac{dh}{dx}dxdh in terms of ttt.
A vertical reinforcement pillar is placed between the ground and the lowest point R R\,R on the track. Find the height of this pillar.
Calculate the acute angle the track makes with the horizontal at the end point QQQ. Give your answer to the nearest degree.
