A man stands on a tree 150 cm high and sees a boat at an angle of depression of 74°. Find the distance of the boat from the base of the tree.

Analyze the question

The tree is 150 cm high, and the angle of depression to the boat is 74°. We need the horizontal distance from the base of the tree to the boat.

Concept

The angle of depression from the top equals the angle of elevation from the boat to the top (alternate angles). This gives a right-angled triangle:

• Opposite side (to the 74° angle) = height of the tree = 150 cm
• Adjacent side = distance d from the base to the boat

So:

tan 74° = opposite / adjacent = 150 / d

Solution

d = 150 / tan 74°

tan 74° ≈ 3.487

d = 150 / 3.487 ≈ 43.0 cm

Answer

Tip

For depression and elevation questions, draw the right-angled triangle first and label the height as "opposite" to the angle at the ground. If the angle is large (like 74°), tan is large, so the horizontal distance must be much smaller than the height. This lets you quickly rule out 52 cm and 15 cm: 150/3.5 is about 43.