2023 RVHS Promo Q11

Fixed vertical cameras have to rotate quickly to keep track of moving objects. A camera at \(A\) is attached on a vertical wall \(AC\) with fixed height \(a\)cm. A ball \(B\) is moving horizontally away from the wall \(AC\). At time \(~t\)seconds, the ball is \(x\) cm from the wall and the camera would have complete a rotation of \(\theta \) radians from the wall to keep track of the ball. At this instant, the camera is rotating at a speed of \(\frac{\mathrm{d}\theta }{\mathrm{d}t}\) radians per second. Assume that the triangle \(ABC\) lies on a vertical plane containing the path of the ball.

1. By expressing \(\tan \theta \) in terms of \(x\) and \(a\), show that \(\frac{\mathrm{d}\theta }{\mathrm{d}t}=\left( \frac{a}{{{a}^{2}}+{{x}^{2}}} \right)\frac{\mathrm{d}x}{\mathrm{d}t}\).[4]
2. When the ball is \(9\) cm from the wall, it is travelling at \(6\) m per second. At this instant, find the speed the camera is rotating at in radians per second, leaving your answer in terms of \(a\).[1]
3. Given that \(a=4\) and that the ball is travelling such that \(x={{t}^{2}}\)cm, find the exact maximum value of the rotating speed of the camera in radians per second.[7]