An airplane is 5,000 m above an observer and 2.1 km to the west of them and 1.5 km to the north of you. Determine the angle to the plane in the x – y axis and the total distance to the plane from you. Choose the x-axis east, y axis north, and z axis up.
distance=sqrt(5^2+2.1^2+1.5^2) km
theta=arcsin(5/distanceabove)
To determine the angle to the plane in the x-y axis and the total distance to the plane from you, we can use trigonometry.
Let's break down the problem step by step:
Step 1: Calculate the distance in the x-axis (west-east direction) between you and the plane.
The plane is 2.1 km to the west of the observer. We assume you are at the origin (0,0) on the x-y plane. Therefore, the distance in the x-axis between you and the plane is 2.1 km.
Step 2: Calculate the distance in the y-axis (north-south direction) between you and the plane.
The plane is 1.5 km to the north of you. Since you are at the origin (0,0) on the x-y plane, the distance in the y-axis between you and the plane is 1.5 km.
Step 3: Calculate the diagonal distance (hypotenuse) between you and the plane.
Using the distance in the x-axis (2.1 km) and the distance in the y-axis(1.5 km), we can use the Pythagorean theorem: a^2 + b^2 = c^2, where a is the distance in the x-axis, b is the distance in the y-axis, and c is the diagonal distance.
So, c^2 = (2.1 km)^2 + (1.5 km)^2
c^2 = 4.41 km^2 + 2.25 km^2
c^2 = 6.66 km^2
c = √6.66 km (taking the square root of both sides)
c ≈ 2.58 km
Step 4: Calculate the angle to the plane in the x-y axis.
To calculate the angle, we can use trigonometric functions. We know the distance in the x-axis (2.1 km) and the diagonal distance (2.58 km), so we can use the cosine function.
Cos(angle) = adjacent / hypotenuse
Cos(angle) = 2.1 km / 2.58 km
Cos(angle) ≈ 0.8134
Using the inverse cosine function or the arc cosine function (acos) on your calculator, you can find the angle. So, angle ≈ acos(0.8134) ≈ 35.83 degrees.
Therefore, the angle to the plane in the x-y axis is approximately 35.83 degrees.
Step 5: Calculate the total distance to the plane from you.
The total distance is the sum of the altitude of the plane (5,000 m) and the diagonal distance (2.58 km = 2,580 m).
Total distance = 5,000 m + 2,580 m = 7580 m
Therefore, the total distance to the plane from you is 7580 meters.