From the roof of the building the angle of depression of the top and foot of the lamp post is 30° and 60° respectively. Find the ratio of the heights of the building and the lamp post.
In ΔEDC, tan 30° = \text{ED}\over \text{DC}
or, 1\over √3 = \text{AE - AD}\over \text{CD} — (1)
In ΔEAB,
tan 60° = \text{AE}\over \text{AB}
or, √3 = \text{AE}\over \text{CD} — (2)
Divide (1) by (2),
\text{AE - AD}\over \text{AE} = 1\over √3 × 1\over √3
or, {\text{AE}\over \text{AE}} - {\text{AD}\over \text{AE}} = 1\over 3
or, 1 – {\text{AD}\over \text{AE}} = 1\over 3
or, {\text{AD}\over \text{AE}} = 1 – 1\over 3
or, {\text{AD}\over \text{AE}} = 2\over 3
or, {\text{BC}\over \text{AE}} = 2\over 3
or, {\text{AE}\over \text{BC}} = 2\over 3
The ratio of the heights of building and lamp post = 3 : 2
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