The angle of elevation of the top of a tower from a point A due south of the tower is $30^\circ$. Walking $40$ m towards the tower to a point B, the angle of elevation changes to $60^\circ$. Find the height of the tower.
Options:
- A: $20$ m
- B: $20\sqrt{3}$ m
- C: $40\sqrt{3}$ m
- D: $10\sqrt{3}$ m
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