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).

By expressing $\theta $ as the difference of two angles, or otherwise, show that
$\tan \theta =\frac{4x}{{{x}^{2}}+4a+{{a}^{2}}}$.
[3]
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.

Explain why a player may decide not to take the kick from the optimal point.
[1]
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]
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]

2020 TYS

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