Choosing a seat in a media room

A media room is \(5.2\text{ m}\) long. A viewer sits at \(X\), a horizontal distance \(d\text{ m}\) from the screen wall.

The viewer's eye is \(1.18\text{ m}\) above the floor and the centre of the screen at \(Y\) is \(1.74\text{ m}\) above the floor.

Two speaker acoustic centres, \(P\) and \(Q\), are at ear height on the back wall and are \(3.2\text{ m}\) apart. The seat lies on the perpendicular bisector of \(PQ\).

Give non-exact numerical answers correct to \(3\) significant figures.

1. Find the angle of elevation from \(X\) to \(Y\) when \(d=1.8\).[2]
2. For comfortable viewing, the angle of elevation must be less than \(14^\circ\). Find the range of suitable values of \(d\).[2]
3. For balanced sound, \(P\), \(Q\) and \(X\) should form an equilateral triangle.
   Find the corresponding value of \(d\), and determine whether it also satisfies the viewing condition.[4]