A motion-activated security sensor at a warehouse entrance sweeps a circular sector OABOABOAB with centre OOO. The sensor has a detection range of r=106 mr = 10\sqrt{6}\text{ m}r=106 m and its field of vision is defined by the central angle AOB=π6AOB = \frac{\pi}{6}AOB=6π radians. A straight security barrier is installed on the floor connecting points AAA and BBB, creating a segment-shaped region between the barrier and the arc ABABAB.
Show that the area of this segment-shaped region is
k(π−3) m2 k(\pi - 3) \text{ m}^2 k(π−3) m2where kkk is an integer to be found.
