A map suggests that Atlanta is 730 miles in a direction 5.00° north of east from Dallas. The same map shows that Chicago is 560 miles in a direction 21.0° west of north from Atlanta. The figure below shows the location of these three cities. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.
I got 621 miles & 70.8 N of E.
Convention:
XY : Cartesian Displacement Vector from X to Y.
Using appropriate capitals for the cities, and the unit vectors î and ĵ for direct East and North.
Dalas to Atlanta is 730 miles 5.00° N of E:
DA = 730 [cos(5.00°) î + sin(5.00°) ĵ]
Atlanta to Chicago 560 miles 21.0° W of N:
AC = 560 [-sin(21.0°) î + cos(21.0°) ĵ]
So from Dalas to Chicago is then:
DC = DA + AC
DC = [(730 cos(5.00°)-560 sin(21.0°)) î + (730 sin(5.00°) + 560 cos(21.0°)) ĵ]
DC ≈ [526.5 î + 586.4 ĵ]
DC ≈ 788.1 [cos(48.08°) î + sin(48.08°) ĵ]
That is, 788.1 miles 48.08° North of East.
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.
To find the displacement from Dallas to Chicago, we can break down the problem into two components: the north-south component and the east-west component.
Step 1: Find the displacement from Dallas to Atlanta
Given that Atlanta is 730 miles in a direction 5.00° north of east from Dallas, we can use the concept of right triangles to find the north-south and east-west components separately.
The north-south component can be calculated using trigonometry:
north-south displacement = 730 miles * sin(5.00°)
The east-west component can be calculated using trigonometry:
east-west displacement = 730 miles * cos(5.00°)
Step 2: Find the displacement from Atlanta to Chicago
Given that Chicago is 560 miles in a direction 21.0° west of north from Atlanta, we can again use trigonometry to find the north-south and east-west components separately.
The north-south component can be calculated using trigonometry:
north-south displacement = 560 miles * cos(21.0°)
The east-west component can be calculated using trigonometry:
east-west displacement = 560 miles * sin(21.0°)
Step 3: Add the components together
To find the displacement from Dallas to Chicago, we add the north-south and east-west components obtained from steps 1 and 2.
North-south displacement = north-south displacement from Dallas to Atlanta + north-south displacement from Atlanta to Chicago
East-west displacement = east-west displacement from Dallas to Atlanta + east-west displacement from Atlanta to Chicago
Finally, we can use the Pythagorean theorem to find the magnitude of the displacement:
Displacement = sqrt((North-south displacement)^2 + (East-west displacement)^2)
By performing these calculations, you should obtain the final displacement from Dallas to Chicago.