2024 SJI P2 Q5

A pony is tied to a corner of a 10 metres by 5 metres rectangular barn, by a rope of length \(x\) metres, where \(5\leq x\leq10\).

Diagram not drawn to scale.

1. Show that the maximum area of the barn that the pony can reach is given by \[\frac{x^2}2\arcsin\left(\frac5x\right)+\frac{5\sqrt{x^2-25}}2\text{ square metres}.\][3]
2. Hence find the maximum length of the rope for the pony to be able to cover at most half the total area of the barn.[3]