2024 ACS(BR) P2 Q10

The diagram shows the side view of a \(14\text{ cm}\) by \(8\text{ cm}\) rectangular block \(PQRS\), placed on a ramp, \(VS\), tilted at an acute angle of \(\theta^\circ\).

The ramp is placed on a horizontal surface \(ST\) and \(d\) is the perpendicular distance from \(Q\) to \(ST\). \(\angle VTS=90^\circ\).

1. Show that \(d=8\cos\theta+14\sin\theta\).[2]
2. Express \(d\) in the form \(R\cos(\theta-\alpha)\), where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[4]
3. Find the smallest value of \(\theta\) such that \(d=10\sqrt2\).[4]