An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 950 m, horizontal distance 19.9 km, and 23.5° south of west. The second aircraft is at altitude 1200 m, horizontal distance 17.4 km, and 22.0° south of west. What is the distance between the two aircraft? (Place the x axis west, the y axis south, and the z axis vertical.)
To find the distance between the two aircraft, we can use the concept of vector addition.
First, let's convert the given information into vector format. We represent a vector by its magnitude (length) and direction. In this case, we have two vectors representing the positions of the aircraft.
For the first aircraft:
Magnitude: 19.9 km
Direction: 23.5° south of west
We can represent this vector as:
Aircraft1 = 19.9 km at 23.5° south of west
For the second aircraft:
Magnitude: 17.4 km
Direction: 22.0° south of west
We can represent this vector as:
Aircraft2 = 17.4 km at 22.0° south of west
Now, we need to find the components of these vectors in the x (west), y (south), and z (vertical) directions.
For Aircraft1:
The horizontal component (x-axis) can be found using cosine function:
Aircraft1_x = 19.9 km * cos(23.5°)
The vertical component (z-axis) is the altitude:
Aircraft1_z = 950 m
For Aircraft2:
The horizontal component (x-axis) can be found using cosine function:
Aircraft2_x = 17.4 km * cos(22.0°)
The vertical component (z-axis) is the altitude:
Aircraft2_z = 1200 m
Now, we can find the distance between the two aircraft using the Pythagorean theorem.
Distance = √((Aircraft2_x - Aircraft1_x)^2 + (Aircraft2_z - Aircraft1_z)^2)
Substituting the values we found:
Distance = √((17.4 km * cos(22.0°) - 19.9 km * cos(23.5°))^2 + (1200 m - 950 m)^2)
Calculating this expression will give us the distance between the two aircraft.