Discussion
From the top of a building 30 m high, the top and bottom of a tower are observed to have angles of depression 30º and 45º respectively. Find the height of the tower.
Answer: D
Let \( d \) be the horizontal distance between the building and the tower.
The tower's foot is 30 m below the observer, at a depression of \( 45^{\circ} \):
\[ d = \frac{30}{\tan 45^{\circ}} = 30 \text{ m} \]
The tower's top, at a depression of \( 30^{\circ} \), is
\[ d\tan 30^{\circ} = \frac{30}{\sqrt{3}} \text{ m} \]
below the building's top, so the tower's height is
\[ 30 - \frac{30}{\sqrt{3}} = 30\left(1 - \frac{1}{\sqrt{3}}\right) \approx 12.7 \text{ m} \]
The tower is shorter than the building, which is consistent with both angles being depressions rather than elevations.
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