A balloon starts from a point P on the ground and rises vertically at a uniform speed. After 6 minutes, an observer standing at a point on level ground 450√3 metres horizontally from P finds that the angle of elevation of the balloon is 60 degrees. What is the speed of the balloon in metres per second?

Introduction / Context:
This is a classic height and distance problem involving a balloon rising vertically at a constant speed. We are given the horizontal distance of the observer from the point where the balloon started, the time taken to reach a certain height, and the angle of elevation at that time. Using trigonometry and the definition of tangent of an angle, we can find the height of the balloon and then compute its vertical speed in metres per second.

Given Data / Assumptions:

• The balloon starts from point P on the ground and rises vertically.
• The observer is on the same horizontal level as P and at a distance of 450√3 metres from P.
• After 6 minutes, the angle of elevation of the balloon from the observer is 60 degrees.
• The balloon rises at a uniform (constant) speed.
• We ignore effects like wind drift and assume the path is perfectly vertical.

Concept / Approach:
When an object is observed at an angle of elevation from a point on the ground, the tangent of that angle equals the vertical height divided by the horizontal distance between the object and the observer. Let the height of the balloon after 6 minutes be h metres. Using tan 60 degrees, we determine h in terms of the known horizontal distance 450√3. Once we have h, we use speed = distance divided by time, with time converted to seconds, to obtain the vertical speed in metres per second.

Step-by-Step Solution:
Step 1: Let h be the height of the balloon after 6 minutes. From the observer, horizontal distance to point P remains 450√3 metres. Step 2: The angle of elevation is 60°, so tan 60° = h / (450√3). Since tan 60° = √3, we have √3 = h / (450√3). Step 3: Multiply both sides by 450√3 to get h = 450√3 * √3 = 450 * 3 = 1350 metres. Step 4: The balloon reaches 1350 metres in 6 minutes, that is 6 × 60 = 360 seconds. Step 5: Speed = distance / time = 1350 / 360 = 3.75 metres per second.

Verification / Alternative check:
Approximate √3 as 1.732. Then horizontal distance is about 779.4 metres. With height 1350 metres, tan of the angle is approximately 1350 / 779.4 which is very close to 1.732, confirming that the angle is 60 degrees. Dividing 1350 by 360 gives 3.75 exactly, so the numerical computation is consistent and accurate.

Why Other Options Are Wrong:
Options 4.25, 4.5, 3.45, and 4.0 metres per second correspond to different heights after 6 minutes, none of which gives tan 60° equal to the ratio of height to 450√3. If we back calculate using these speeds, we do not recover a consistent angle of 60 degrees with the given horizontal distance, so they are incorrect.

Common Pitfalls:
A common error is to misinterpret the 450√3 metres as the height instead of the horizontal distance. Another mistake is to forget to convert 6 minutes into seconds when computing metres per second. Some learners also confuse sine and tangent, but in this type of problem, tangent is the correct ratio because it connects opposite side and adjacent side of the right angled triangle.

Final Answer:
The speed of the balloon is 3.75 metres per second.