A man standing on the top of a 120 m high tower observes the angles of depression of two cars on opposite sides of the tower to be 30°and 45°. What is the distance between the two cars?

Correct — D. The angle of depression from the top equals the angle of elevation from each car (alternate angles), so each car's distance = height ÷ tan(angle).

Car at 45°: 120 ÷ tan 45° = 120 ÷ 1 = 120 m
Car at 30°: 120 ÷ tan 30° = 120 × √3 = 120√3 m

Opposite sides, so add the two distances:
120√3 + 120 = 120(√3 + 1) m → option (d)

Why the others are wrong

• (a)
  120(√3 − 1) m — 120(√3 − 1) m is the gap when both cars stand on the same side of the tower. Here they are on opposite sides, so the two distances add.
• (b)
  120√3 m — 120√3 m is only the distance of the car seen at 30°. It leaves out the 120 m to the car on the other side.
• (c)
  120 (1 + 1⁄√3) m — 120(1 + 1⁄√3) m multiplies by tan 30° instead of dividing. The 30° car is the farther one, so its distance must exceed 120 m. 120⁄√3 ≈ 69 m does not, while 120√3 ≈ 208 m does.

The angle of depression is measured down from the horizontal at the observer's eye. That horizontal is parallel to the ground, so it equals the angle of elevation of the observer seen from the object.

In the right triangle of tower, ground and line of sight, tan θ = height ÷ ground distance, so distance = height ÷ tan θ = height × cot θ.

A smaller angle puts the object farther away.

Key facts

• tan 30° = 1⁄√3, tan 45° = 1, tan 60° = √3.
• The angle of depression from the top of a tower equals the angle of elevation of the tower's top from the object (alternate angles).
• cot 30° = √3 ≈ 1.732, so an object seen at 30° from a height h is h√3 away along the ground.

Common traps

• Subtracting the two distances by habit. That is the same-side case and gives 120(√3 − 1) m.
• Multiplying the height by tan θ instead of dividing, which puts the 30° car nearer than the 45° car.

No directly related past PYQ was found.