2019 NYJC P1 Q11

A boy is playing a ball game on a field. He arranges two cones \(A\) and \(B\) along the end of the field such that the cones are \(~a\) and \(b\) metres respectively from one corner, \(O\), of the field as shown in the diagram below. The boy stands along the edge of the field at \(x\) metres from \(O\) and kicks the ball between the two cones. The angle that the two cones subtends at the position of the boy is denoted by \(\theta \).

1. Show that \(\tan \theta =\left( b-a \right)\frac{x}{{{x}^{2}}+ab}\).[2]
2. It is given that \(a=15\) and \(b=20\). Find by differentiation, the value of \(x\) such that \(\theta \) is at a maximum.[3]
3. It is given instead that the boy gets two friends to vary the position of both cones \(A\) and \(B\) along the end of the field such that \(5\le a\le 12\) and \(b=2a\), and the boy moves along the edge of the field such that his distance from cone \(A\) remains unchanged at \(18\) metres. Sketch a graph that shows how \(\theta \) varies with \(a\) and find the largest possible value of \(\theta \).[4]
4. The boy runs until he is at a distance \(k\) metres from the goal line that is formed by the two cones and kicks the ball toward the goal line. The path of the ball is modelled by the equation \(h=-{{\left( \frac{1}{10}k+2 \right)}^{2}}+6\), where \(k\) is the distance of the ball from the goal line and \(h\) its corresponding height above the ground respectively. Find the angle that the path of the ball makes with the horizontal at the instant the ball crosses the goal line.[3]