A boat is moving away from a 150 m high cliff. From the top of the cliff, the angle of depression of the boat changes from 60� to 45� in 2 minutes. Find the speed of the boat in metres per hour (m/h).

Solution -

Step 1: Given Data

• Height of the cliff = 150 m
• Initial angle of depression = 60°
• Final angle of depression = 45°
• Time taken = 2 minutes

Let the distance of the boat from the cliff initially be x and later be y.

Step 2: Distance when the angle of depression is 60°

Using the trigonometric relation:

tan θ = Height / Distance

tan 60° = 150 / x

√3 = 150 / x

x = 150 / √3

x = 50√3 m

So, the initial distance of the boat from the cliff is 50√3 m.

Step 3: Distance when the angle of depression is 45°

tan 45° = 150 / y

1 = 150 / y

y = 150 m

So, the second distance of the boat from the cliff is 150 m.

Step 4: Distance travelled by the boat

Distance travelled = y − x

= 150 − 50√3

≈ 150 − 86.6

≈ 63.4 m

The boat travelled approximately 63.4 metres in 2 minutes.

Step 5: Calculate Speed

Convert time into hours:

2 minutes = 2/60 = 1/30 hour

Speed = Distance / Time

Speed = 63.4 ÷ (1/30)

Speed = 63.4 × 30

≈ 1902 m/h

Final Answer - 

The speed of the boat is approximately 1902 metres per hour.