Mountain peaks on a scale drawing

Core stretch

A landscape is drawn to scale \(1:200\,000\), so 1 cm represents 2 km. Points \(A\) and \(B\) are mountain peaks.

1. Their horizontal separation \(AC\) is 4 cm on the drawing; find the actual separation.[2]
2. Their actual heights are 3.2 km and 8.2 km; find \(\angle BAC\).[2]
3. From ground point \(P\), \(\angle APQ=12^\circ\); find \(\angle BPQ\).[2]
4. A lake has actual area \(32\text{ km}^2\); find its drawing area. [2]

Give non-exact lengths and areas to 3 significant figures, and angles to 1 decimal place.