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An air traffic controller notices two signals from two planes on the radar monitor. One plane is at 10.9-km horizontal distance to the tower in a direction 33.9° south of west. The second plane is at altitude of 5601 m a is 10.7 km directed 21.9° south of west. What is the distance between these planes in kilometers? (25%) Problem 4: An air traffic controller notices two signals from two planes on the radar monitor. One plane is at altitude 1123 m and a 10.9-km horizontal distance to the tower in a direction 33.9* south of west. The second plane is at altitude of 5601 m and its horizontal distance is 10.7 km directed 21.9south of west. What is the distance between these planes in kilometers?

1 Answer

4 votes

Answer:

about 5.019 km

Step-by-step explanation:

You want the distance between two airplanes whose locations are ...

  • 10.9 km W 33.9° S at 1123 m altitude
  • 10.7 km W 21.9° S at 5601 m altitude

Coordinates

In (-x, -y, z) coordinates, the coordinates of the two airplanes in km are ...

Plane 1: (10.9·cos(33.9°), 10.9·sin(33.9°), 1.123) ≈ (9.047, 6.079, 1.123)

Plane 2: (10.7·cos(21.9°), 10.7·sin(21.9°), 5.601) ≈ (9.928, 3.991, 5.601)

The difference between these coordinates is ...

Plane 2 - Plane 1 = (9.928, 3.991, 5.601) -(9.047, 6.079, 1.123)

Plane 2 -Plane 1 = (0.881, -2.088, 4.478)

Distance

The distance between the planes is the root of the sum of the squares of these distances:

d = √(0.881² +(-2.088)² +4.478²) ≈ 5.019 . . . . km

The distance between the planes is about 5.019 km.

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Additional comment

We can use any convenient consistent coordinate system. Here, since both angles are measured CCW from west, we choose a coordinate system with the x-y plane rotated 180° from the usual position. The distances and relative angles are unchanged by this rotation.

Since we want the distance between planes at different altitudes, we have cast the problem as a 3D problem with axis measures in kilometers. Then the usual procedure for finding distances between points applies.

We have assumed both distance and angle measures are from the tower to the airplane. The wording of the problem is somewhat ambiguous. If one of these measures is from the airplane to the tower, then the distance between planes will be quite different.

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An air traffic controller notices two signals from two planes on the radar monitor-example-1
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