A balloon is connected to a meteorological ground station by a cable of length $100$ m, inclined at $30^\circ$ to the horizontal. Assuming there is no slack in the cable, determine the height of the balloon from the ground.
Options:
- A: $50\sqrt{3}$ m
- B: $50$ m
- C: $100\sqrt{3}$ m
- D: $100$ m
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