The angle of elevation and depression of the top and bottom of another building, measured from the top of a 24…

Analyze the question

You stand on top of a 24 m building. Looking at the top of a second building, the angle of elevation is 30°. Looking at its bottom, the angle of depression is 60°. We need the second building's total height.

Concept

Use right-angled triangle trigonometry (tan = opposite/adjacent). Let d be the horizontal distance between the two buildings.

Solution

1. Angle of depression to the bottom is 60°. The vertical drop is the height of the first building, 24 m: tan 60° = 24/d √3 = 24/d d = 24/√3 = 8√3 m

2. Angle of elevation to the top is 30°. The extra height h above the observer's eye level is: tan 30° = h/d h = d × tan 30° = 8√3 × (1/√3) = 8 m

3. Total height of the second building: 24 + 8 = 32 m

Answer

The second building is 32 m tall, which is option D.

Tip

In "two buildings" problems, the angle of depression to the bottom gives you the horizontal distance, because the first building's height is the known opposite side. Then use the angle of elevation to find the extra height above your eye level, and add the two parts. Remember that tan 60° = √3 and tan 30° = 1/√3.