An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 750 m, horizontal distance 19.8 km, and 22.5° south of west. The second aircraft is at altitude 1000 m, horizontal distance 18.0 km, and 18.0° 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.)

Question

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 750 m, horizontal distance 19.8 km, and 22.5° south of west. The second aircraft is at altitude 1000 m, horizontal distance 18.0 km, and 18.0° 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.)

Answer 1

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 750 m, horizontal distance 19.8 km, and 22.5° south of west. The second aircraft is at altitude 1000 m, horizontal distance 18.0 km, and 18.0° 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.)

Answer 2

To find the distance between the two aircraft, we can use the three-dimensional distance formula:

Distance = √[(Δx)^2 + (Δy)^2 + (Δz)^2]

First, let's calculate the Δx (change in x-coordinate) between the two aircraft.

For the first aircraft:
Horizontal distance = 19.8 km
Angle south of west = 22.5°

To calculate the x-coordinate:
x-coordinate = horizontal distance * cos(angle)
x1 = 19.8 km * cos(22.5°)

Next, let's calculate the Δy (change in y-coordinate) between the two aircraft.

For the first aircraft:
Horizontal distance = 19.8 km
Angle south of west = 22.5°

To calculate the y-coordinate:
y-coordinate = horizontal distance * sin(angle)
y1 = 19.8 km * sin(22.5°)

Now, let's calculate the Δz (change in z-coordinate) between the two aircraft.

For the first aircraft:
Altitude = 750 m

Now that we have the Δx, Δy, and Δz for the first aircraft, we can repeat the same process for the second aircraft and find the Δx2, Δy2, and Δz2.

For the second aircraft:
Horizontal distance = 18.0 km
Angle south of west = 18.0°

To calculate the x-coordinate:
x-coordinate = horizontal distance * cos(angle)
x2 = 18.0 km * cos(18.0°)

To calculate the y-coordinate:
y-coordinate = horizontal distance * sin(angle)
y2 = 18.0 km * sin(18.0°)

For the second aircraft:
Altitude = 1000 m

Now, we can substitute these values into the three-dimensional distance formula:

Distance = √[(Δx)^2 + (Δy)^2 + (Δz)^2]

Distance = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]

Where:
Δx = x2 - x1
Δy = y2 - y1
Δz = z2 - z1

Substituting the values into the equation, we can find the distance between the two aircraft.