2017 JJC P1 Q10

A laser from a fixed point \(O\) on a flat ground projects light beams to the top of two vertical structures \(A\) and \(B\) as shown above. To project the beam to the top of \(A\), the laser makes an angle of elevation of \(\frac{\pi }{6}\) radians. To project the beam to the top of \(B\), the laser makes an angle of elevation of \(\left( \frac{\pi }{6}+x \right)\) radians. The two structures \(A\) and \(B\) are of heights \(h\) m and \(\left( h+\sqrt{3}k \right)\)m respectively and are \(10\)m and \(\left( 10+k \right)\)m away from \(O\) respectively.

1. Show that the length of the straight beam from \(O\) to \(A\) is \(\frac{20}{\sqrt{3}}\)m.[1]
2. Show that the length of \(AB\) is \(2k\)m and that the angle of elevation of \(B\) from \(A\) is \(\frac{\pi }{3}\) radians.[3]
3. Hence, using the sine rule, show that \(k=\frac{10\sin x}{\sqrt{3}\sin \left( \frac{\pi }{6}-x \right)}\).[2]
4. If \(x\) is sufficiently small, show that \(k\approx \frac{20}{\sqrt{3}}\left( x+a{{x}^{2}} \right)\), where \(a\) is a constant to be determined.[6]