A boat moves directly away from the foot of an observation tower. At one instant, the angle of depression of the boat from the observer's eye at the top of the tower is 60° when the boat is 50 m from the tower. After 8 seconds, the angle of depression becomes 30°. Assuming still water and straight-line motion, what is the approximate speed of the boat in km/h?

Introduction / Context:
This question tests your understanding of angles of depression and how to convert a geometric change in position into a speed in km/h. The observer stands at the top of a tower, and as the boat moves away, the angle of depression decreases from 60° to 30° over a known time interval.

Given Data / Assumptions:

• Angle of depression to the boat initially = 60°.
• Horizontal distance of the boat from the tower at that instant = 50 m.
• Angle of depression after 8 seconds = 30°.
• Boat moves in a straight horizontal line directly away from the tower.
• The height of the tower remains constant.
• We assume no currents and treat water as still, so motion is uniform.

Concept / Approach:
An angle of depression from the top of a tower equals the angle of elevation from the boat to the top of the tower. Using basic trigonometry, tan(theta) = opposite / adjacent, where the opposite side is the tower height and the adjacent side is the horizontal distance. We can find the height of the tower from the first observation, use it to determine the second distance, then compute how far the boat travelled in 8 seconds and convert that to km/h.

Step-by-Step Solution:
Let h be the height of the tower.Initially, tan(60°) = h / 50.tan(60°) = √3, so h = 50 * √3.After 8 s, angle of depression = 30°, so tan(30°) = h / d, where d is new distance.tan(30°) = 1 / √3, hence 1 / √3 = (50 * √3) / d.So, d = 150 m.Distance travelled in 8 s = d - 50 = 150 - 50 = 100 m.Speed in m/s = 100 / 8 = 12.5 m/s.Convert to km/h: speed = 12.5 * 3.6 = 45 km/h.

Verification / Alternative check:
If the speed is 45 km/h, then per second the boat travels 45 * 1000 / 3600 = 12.5 m/s, confirming the displacement of 100 m in 8 seconds. The resulting distances and angles are consistent with the tangent values for 60° and 30°.

Why Other Options Are Wrong:
33 km/h and 36 km/h: These represent smaller speeds and would not allow the boat to move 100 m in just 8 seconds.42 km/h: Close but slightly low; gives less than 100 m in 8 seconds.50 km/h: Slightly high; would imply a greater horizontal displacement than allowed by the angle change.

Common Pitfalls:
Common mistakes include using the sine instead of the tangent function, forgetting that angle of depression equals angle of elevation, or failing to convert from m/s to km/h correctly (multiplying by 3.6). Another pitfall is forgetting to subtract the initial distance to find the distance actually travelled in the 8 seconds.

Final Answer:
The approximate speed of the boat is 45 km/h.