N2020 P1 Q11

In sport science, studies are made to optimise performance in all aspects of sport from fitness to technique.

In a game, a player scores \(3\) points by carrying the ball over the scoring line, shown in the diagram as \(XY\). When a player has scored these \(3\) points, an extra point is scored if the ball is kicked between two fixed vertical posts at \(C\) and \(D\). The kick can be taken from any point on the line \(AB\), where \(A\) is the point at which the player crossed the scoring line and \(AB\) is perpendicular to \(XY\).

The distance \(CD\) is \(4\)m; \(XC\) is equal to \(DY\); the point \(A\) is a distance \(a\) m from \(C\) and \(A\) lies between \(X\) and \(C\). The kick is taken from the point \(K\), where \(AK\) is \(x\)m. The angle \(CKD\) is \(\theta \) (see diagram).

1. By expressing \(\theta \) as the difference of two angles, or otherwise, show that
   \(\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}\).[3]
2. Find, in terms of \(a\), the value of \(x\) which maximises \(\tan \theta \), simplifying your answer. Find also the corresponding value of \(\tan \theta \). (You need not show that your answer gives a maximum.)[3]

The point corresponding to the value of \(x\) found in part (ii) is called the optimal point. The corresponding value of \(\theta \) is called the optimal angle.

3. Explain why a player may decide not to take the kick from the optimal point.[1]
4. Show that, when \(\theta \) is the optimal angle, \(\tan KDA=\sqrt{\frac{a}{4+a}}\). Find the approximate value of angle \(KDA\) when \(a\) is much greater than \(4\).[3]
5. It is given that the length of the scoring line \(XY\) is \(50\)m. Find the range in which the optimal angle lies as the location of \(A\) varies between \(X\) and \(C\).[2]