That is fire station Q is at a distance of 6.34 km from the building and fire station P is at a distance 3.66 km.Hence fire station P has to send his team first and it has to travel a distance of 3.66 km

Analysis of the situation:

To determine which fire station should send its team and the distance they would have to travel, we need to analyze the given information and use basic trigonometry.

Given information:
- The two fire stations, P and Q, are 20 km apart from each other on a straight road.
- P observes that the fire is at an angle of 60 degrees to the road.
- Q observes that the fire is at an angle of 45 degrees to the road.

Approach:
1. Determine the location of the fire using the angles observed by the two fire stations.
2. Calculate the distances of the fire from each fire station using trigonometry.
3. Compare the distances and decide which fire station should send its team.

Calculations:

Step 1: Determine the location of the fire
- Draw a diagram to represent the situation.
- Place the two fire stations, P and Q, 20 km apart on a straight road.
- Mark the angles of 60 degrees and 45 degrees at the fire stations, according to their observations.

Step 2: Calculate the distances of the fire from each fire station
- Let the distance from station P to the fire be x km.
- Let the distance from station Q to the fire be y km.
- Use trigonometry to calculate x and y.

For station P:
- In the triangle formed by station P, the fire, and the point on the road directly opposite the fire, we have a right triangle.
- The angle between the road and the line connecting station P and the fire is 60 degrees.
- Therefore, we can use the sine function to calculate x:
sin(60) = x / 20
x = 20 * sin(60)
x = 20 * √3 / 2
x = 10√3 km

For station Q:
- In the triangle formed by station Q, the fire, and the point on the road directly opposite the fire, we have a right triangle.
- The angle between the road and the line connecting station Q and the fire is 45 degrees.
- Therefore, we can use the sine function to calculate y:
sin(45) = y / 20
y = 20 * sin(45)
y = 20 * √2 / 2
y = 10√2 km

Step 3: Compare the distances and decide which fire station should send its team
- The distance from station P to the fire, x, is 10√3 km.
- The distance from station Q to the fire, y, is 10√2 km.
- Comparing the two distances, we find that x > y.
- Therefore, station Q should send its team as they have to travel a shorter distance.

Conclusion:

Based on the given information and calculations, it is recommended that fire station Q should send its team to the building fire. The team from station Q would have to travel a distance of 10√2 km to reach the building fire.