The parametric equations
model the motion of a roller coaster train along a section of the track as time \(t\) varies. The variable \(x\) represents the horizontal distance travelled by the train from its starting point, and \(y\) represents the vertical distance from the ground. In this question, distance is measured in metres and time is measured in seconds.
The speed of the roller coaster train at any time \(t\) is given by the formula[2]
\(\sqrt{\left(\frac{dy}{dt}\right)^2+\left(\frac{dx}{dt}\right)^2}.\)
Calculate the speed of the train when \(x=9\), giving your answer correct to three decimal places.
Use differentiation to find the coordinates of the turning point in this section of the track and prove that it is a maximum point.[4]
A designer is creating a wind-blocking panel to be attached beneath the roller coaster track. Its cross-sectional area is modelled by the area under the parametric curve from \(t=0\) to \(t=6\). The area of the region under a parametric curve from \(t=\alpha\) to \(t=\beta\) is given by the formula
\(\int_{\alpha}^{\beta} y\left(\frac{dx}{dt}\right)\,dt.\)
Show that the cross-sectional area of the panel is \(\int_a^b kt^2 e^{-t}\,dt\), where \(a\), \(b\) and \(k\) are integers to be determined.[2]
Find the exact area of the panel.[4]