A balloon is rising vertically upwards. An an instant, an observation on the ground, whose distance from the balloon is 100 meters, sees the balloon at an angle of elevation of `30^(@)`. If the balloon rises further vertically to a point where the angle of elevation as seen by the observer is `45^(@)`, then its height (in meters) from the ground is
`("Take "sqrt3=1.73)`

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understanding the situation
We have an observer on the ground who sees a balloon at an angle of elevation of \(30^\circ\) when it is 100 meters away horizontally. We need to find the height of the balloon when the angle of elevation changes to \(45^\circ\).

### Step 2: Setting up the first triangle
Let’s denote:
- The height of the balloon at the first observation as \(h_1\).
- The horizontal distance from the observer to the balloon as \(d = 100\) meters.

Using the angle of elevation \(30^\circ\):
\[ \tan(30^\circ) = \frac{h_1}{d} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h_1}{100} \]
Thus,
\[ h_1 = \frac{100}{\sqrt{3}} = \frac{100 \times 1.73}{3} \approx 57.67 \text{ meters} \]

### Step 3: Setting up the second triangle
Now, when the balloon rises further, the angle of elevation becomes \(45^\circ\). Let the new height of the balloon be \(h_2\).

Using the angle of elevation \(45^\circ\):
\[ \tan(45^\circ) = \frac{h_2}{d} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{h_2}{100} \]
Thus,
\[ h_2 = 100 \text{ meters} \]

### Step 4: Finding the total height from the ground
Now, we need to find the total height of the balloon from the ground when it is at the \(45^\circ\) angle of elevation.

The height of the balloon when it was first observed was \(h_1 \approx 57.67\) meters. The balloon has risen from this height to \(h_2 = 100\) meters.

### Step 5: Conclusion
The total height of the balloon from the ground when the angle of elevation is \(45^\circ\) is:
\[ h_2 = 100 \text{ meters} \]

### Final Answer
The height of the balloon from the ground is **100 meters**.

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