To find the displacement from Dallas to Chicago, we can break down the information given step-by-step:
Step 1: Find the components of the Atlanta to Dallas displacement.
From the given information, the distance from Atlanta to Dallas is 730 miles in a direction 5.00° north of east.
To find the horizontal component, we multiply the distance by the cosine of the angle:
Horizontal component = 730 miles * cos(5.00°)
To find the vertical component, we multiply the distance by the sine of the angle:
Vertical component = 730 miles * sin(5.00°)
Step 2: Find the components of the Chicago to Atlanta displacement.
From the given information, the distance from Chicago to Atlanta is 560 miles in a direction 21.0° west of north.
To find the horizontal component, we multiply the distance by the sine of the angle:
Horizontal component = 560 miles * sin(21.0°)
To find the vertical component, we multiply the distance by the cosine of the angle:
Vertical component = 560 miles * cos(21.0°)
Step 3: Combine the components to find the displacement from Dallas to Chicago.
To find the displacement from Dallas to Chicago, we need to subtract the components of the Atlanta to Dallas displacement from the components of the Chicago to Atlanta displacement.
Horizontal displacement = Horizontal component of Chicago to Atlanta - Horizontal component of Atlanta to Dallas
Vertical displacement = Vertical component of Atlanta to Dallas + Vertical component of Chicago to Atlanta
To find the overall displacement, we use the Pythagorean theorem:
Displacement = √(Horizontal displacement^2 + Vertical displacement^2)
By performing the calculations, the resulting displacement is approximately 621 miles and the direction is 70.8° north of east. Therefore, your answer of 621 miles and 70.8° N of E is correct.