2020 ACJC P1 Q7

A hot air balloon company gives a sticker to each passenger who is going for a ride. The sticker is in the shape of a curve \(C\) given by the parametric equations

\(x=2\cos \theta +\sin 2\theta ,\,\,y=2+2\sin \theta ,\)for \(-\pi <\theta <\pi \).

The curve has a \(y\)- intercept at \((0,4)\).

1. Use a non-calculator method to find the exact area of the sticker in terms of \(\pi \).[5]
2. Prove that \(x=y\cos \theta \). Deduce that the Cartesian equation of the curve \(C\) is
   \(4{{x}^{2}}=4{{y}^{3}}-{{y}^{4}}\).[3]
3. A model of a hot air balloon is made by rotating the curve \(C\) about the \(y\)- axis through \(\pi \) radians. Find the volume of the model.[2]