2025 CGSS PRELIMS P2 Q9

Ryan has set up a home theatre system in one of his rooms, which is in the shape of a cuboid with dimensions \(4\text{ m}\) by \(3.6\text{ m}\) by \(3\text{ m}\). Figure 1 shows the three-dimensional layout of the room, where a television is mounted on the front wall \(QRVU\) and two speakers are mounted on the back wall \(PSWT\), at \(P\) and \(T\) respectively.

Ryan is planning to put an armchair along \(MN\), which is a straight path that cuts across the centre of the room from the back to the front. Figure 2 shows the cross-sectional view of him sitting in the armchair, where his eye level at \(X\) is \(1.24\text{ m}\) directly above the ground and \(d\text{ m}\) away from the front wall. The centre of the television at \(Y\) is \(1.6\text{ m}\) directly above \(N\).

1. Find the angle of elevation from \(X\) to \(Y\) when \(d=1.2\text{ m}\).[2]
2. Ryan found out that, to have an optimal view of the television, the angle of elevation from \(X\) to \(Y\) should be less than \(15^\circ\), and to have an optimal audio experience, the position of the two speakers and \(X\) should form an equilateral triangle. Suggest a suitable value for \(d\) such that the position of the armchair will provide Ryan with the best possible user experience. Justify the decisions you make and show your calculations clearly.[7]