A balloon leaves the earth and rises at a uniform velocity. At the end of 2 min, an observer situated at 200m from the point the balloon was released, finds the regular elevation of the balloon to be `60^(@)`. What is the speed in meters per second, to the nearest integer, of the balloon?

Answer

To find the speed of the balloon, we will follow these steps:

### Step 1: Understand the Problem
We know that a balloon is rising at a uniform velocity, and after 2 minutes, the observer sees the balloon at an angle of elevation of \(60^\circ\) from a point that is 200 meters away horizontally from the release point.

### Step 2: Set Up the Right Triangle
The situation can be visualized as a right triangle where:
- The height of the balloon (h) is the vertical side.
- The horizontal distance from the observer to the point of release is 200 meters (the base).
- The angle of elevation is \(60^\circ\).

### Step 3: Use the Tangent Function
Using the tangent function, we can relate the height of the balloon to the horizontal distance:
\[ \tan(60^\circ) = \frac{\text{height}}{\text{horizontal distance}} = \frac{h}{200} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we can write:
\[ \sqrt{3} = \frac{h}{200} \]

### Step 4: Solve for Height (h)
Rearranging the equation to solve for h:
\[ h = 200 \cdot \sqrt{3} \]

### Step 5: Calculate Height
Now, we can calculate the height:
\[ h = 200 \cdot \sqrt{3} \approx 200 \cdot 1.732 \approx 346.4 \text{ meters} \]

### Step 6: Convert Time to Seconds
The time taken for the balloon to rise is given as 2 minutes. We need to convert this into seconds:
\[ 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]

### Step 7: Calculate Speed
Speed is defined as distance traveled over time taken. Therefore, the speed of the balloon can be calculated as:
\[ \text{Speed} = \frac{\text{Height}}{\text{Time}} = \frac{346.4}{120} \approx 2.8867 \text{ m/s} \]

### Step 8: Round to the Nearest Integer
Finally, rounding \(2.8867\) to the nearest integer gives us:
\[ \text{Speed} \approx 3 \text{ m/s} \]

### Final Answer
The speed of the balloon is approximately **3 m/s**.

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