N2025 P1 Q11

Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations

\(x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),\)

for \(-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), where \(x\) and \(y\) are measured in metres. The \(x\)-axis models the horizontal ground.

The maximum point on the curve is at \(A\), and the line \(AB\) is vertical. Points \(C\) and \(D\) are where the tangent to the curve is parallel to the \(y\)-axis. Point \(E\) is the intersection of the lines \(AB\) and \(CD\).

1. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta\).[3]
2. 1. Find the values of \(\theta\) at points \(C\) and \(D\), giving your answers correct to \(2\) significant figures.[3]
   2. Hence find the greatest width, \(CD\), of the loop.[2]

The \(x\)-coordinate of point \(A\) is \(28\).

3. 1. Find the value of \(\theta\) at point \(A\).[1]
   2. Find the \(2\) possible values of \(\theta\) at point \(B\).[2]
   3. Hence find the distance \(AB\).[1]

Engineers know that, for safety reasons, the distance \(AE\) should be within \(10\%\) of the distance \(\frac{1}{2}CD\).

4. Determine whether the design of this section of track satisfies this condition.[2]