ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are ${\cot }^{-1}\left(3\sqrt{2}\right)$ and  $\cos e{c}^{-1}\left(2\sqrt{2}\right)$respectively, then the height of the tower (in metres) is :