2020 SJI P2 Q3

Farmer Adam has a fenced rectangular field, measuring \(10\) metres by \(4\) metres, for his goat, Billy, to graze. To keep Billy from running away, Adam attaches a rope to a wooden post at one corner of his field, and the other end of the rope to Billy. The length of Billy’s rope, \(x\) metres, where \(4<x<10\), is such that Billy can graze an area that is exactly \(\dfrac34\) of Adam’s field.

1. Show that \(x\) satisfies the equation \[x^2\arcsin\left(\frac4x\right)+4\sqrt{x^2-16}=60.\][5]
2. Find the value of \(x\).[2]