A balloon is connected to a meteorological station by a cable of length 200 m inclined to the horizontal at an angle of 60°. Determine the height of the balloon from the ground. Assume that there is no slack in a cable. [Take `sqrt3 = 1.73)`.

Answer

To determine the height of the balloon from the ground, we can use trigonometry. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a right triangle formed by the cable, the height of the balloon, and the horizontal distance from the meteorological station to the point directly below the balloon. The cable length is the hypotenuse, and the height of the balloon is the opposite side.

### Step 2: Identify the Given Values
- Length of the cable (hypotenuse) = 200 m
- Angle of inclination (angle with the horizontal) = 60°

### Step 3: Set Up the Trigonometric Relationship
In a right triangle, we can use the sine function to relate the angle to the opposite side (height of the balloon) and the hypotenuse (length of the cable):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, \(\theta = 60^\circ\), opposite = height (h), and hypotenuse = 200 m.

### Step 4: Substitute the Values into the Equation
\[ \sin(60^\circ) = \frac{h}{200} \]

### Step 5: Use the Value of \(\sin(60^\circ)\)
The value of \(\sin(60^\circ)\) is \(\frac{\sqrt{3}}{2}\). Therefore, we can rewrite the equation as:
\[ \frac{\sqrt{3}}{2} = \frac{h}{200} \]

### Step 6: Solve for h
To find h, we can rearrange the equation:
\[ h = 200 \times \frac{\sqrt{3}}{2} \]
\[ h = 100\sqrt{3} \]

### Step 7: Substitute the Value of \(\sqrt{3}\)
Given that \(\sqrt{3} \approx 1.73\):
\[ h = 100 \times 1.73 = 173 \text{ m} \]

### Final Answer
The height of the balloon from the ground is **173 meters**.

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