Question

Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. A guard, stationed at the top of a 240 m tower, observed an unidentified boat coming towards it. A clinometer or inclinometer is an instrument used for measuring angles or slopes(tilt). The guard used the clinometer to measure the angle of depression of the boat coming towards the lighthouse and found it to be 30°.

1. Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
2. After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?

Answer 1

i. 

In ∆PTR, tan 30° = `240/x` ⇒ x = `240sqrt(3)` m

ii. Distance of boat from tower = `240sqrt(3) - 240(sqrt(3) - 1)` = 240 m

Let the angle of depression = θ

tan θ = `240/240` = 1 ⇒ θ = 45°