From the top of a building 78 m high, the angles of depression of the bottom and the top of a tower are observed to be 45° and 30° respectively. Determine the approximate height of the tower so observed?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 33 m
An angle of depression is the angle between the horizontal line through the observer’s eye and the line of sight to a point below. Each angle sits in a right triangle, whose sides are linked by the tangent: \(\tan\theta=\frac{\text{opposite side}}{\text{adjacent side}}\).
In triangle ABC, \(\tan45^\circ=\frac{AB}{BC}=1\), so BC \(=\) AB \(=78\) m: the tower stands 78 m from the building
ED is level with the top of the tower, so ED \(=\) BC \(=78\) m
In triangle AED, \(\tan30^\circ=\frac{AE}{ED}\), so AE \(\displaystyle =\frac{78}{\sqrt{3}}=26\sqrt{3}\approx45.03\) m
Height of the tower \(h=\) EB \(=78-45.03=32.97\approx33\) m