Why the Formulas Work · Superchallenge problem

From a point $P$ on level ground, the angle of elevation of the top of a vertical tower is $\theta$. After walking $30$ m straight toward the tower, to $Q$, the angle of elevation is $2\theta$. After walking a further $10\sqrt3$ m toward the tower, to $R$, it is $4\theta$. Find the height of the tower, in metres.