A person standing at point P sees angle of elevation of the top of a building, whose base is 50 meters away, to be `60^(@)`. Another building whose base is 20 meters away from the base of the first building and is between the observer and first building has height h meters, then the maximum possible height (in meters) of this second building is `("Take "sqrt3=1.73)`

Answer

To solve the problem step by step, we will use trigonometric principles.

### Step 1: Understand the scenario
We have two buildings. The first building is 50 meters away from the observer at point P, and the angle of elevation to the top of this building is 60 degrees. The second building is 20 meters away from the base of the first building, making it 30 meters away from the observer.

### Step 2: Find the height of the first building
Using the angle of elevation and the distance from the observer, we can find the height of the first building (let's call it \( H_1 \)).
- The distance from point P to the first building is 50 meters.
- The angle of elevation to the first building is 60 degrees.

Using the tangent function:
\[ \tan(60^\circ) = \frac{H_1}{50} \]
We know that \( \tan(60^\circ) = \sqrt{3} \).

So, substituting this into the equation:
\[ \sqrt{3} = \frac{H_1}{50} \]
Now, solving for \( H_1 \):
\[ H_1 = 50 \cdot \sqrt{3} \]

### Step 3: Calculate the height of the second building
The second building is 20 meters away from the first building, making it 30 meters away from the observer. We need to find the maximum possible height \( h \) of the second building such that it can still be seen from point P.

Using the same angle of elevation (60 degrees) for the second building:
\[ \tan(60^\circ) = \frac{h}{30} \]
Again, substituting \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h}{30} \]
Now, solving for \( h \):
\[ h = 30 \cdot \sqrt{3} \]

### Step 4: Substitute the value of \( \sqrt{3} \)
We are given that \( \sqrt{3} = 1.73 \). Therefore:
\[ h = 30 \cdot 1.73 \]
Calculating this gives:
\[ h = 51.9 \text{ meters} \]

### Conclusion
The maximum possible height of the second building is \( 51.9 \) meters.

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