A stationary boat is observed from a height of 100m. If the horizontal distance between the observer and the…

Analyze the question

An observer is 100 m above the water, and the boat is 80 m away horizontally. The angle of depression is measured downward from the observer's horizontal line of sight to the boat.

Concept

The angle of depression equals the angle of elevation from the boat to the observer (alternate angles). In the right-angled triangle:

• Opposite side (height) = 100 m
• Adjacent side (horizontal distance) = 80 m

tan θ = opposite / adjacent

Solution

tan θ = 100/80 = 1.25

θ = tan⁻¹(1.25) ≈ 51.34°

Check: tan 51.34° ≈ 1.25, which is correct. (The 53.13° option would come from tan θ = 4/3, and 36.87° from tan θ = 0.75, which are the wrong ratios.)

Answer

The angle of depression is 51.34°, which is option C.

Tip

For angles of depression, draw the horizontal line from the observer and the angle sits below it. Then use the alternate angle at the boat. Always match "opposite" to the vertical height and "adjacent" to the horizontal distance. If you divide height by distance, you get tan θ. A quick check: since the height is greater than the distance, tan θ > 1, so the angle must be greater than 45°. That immediately eliminates A and B.