Question

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.

magnitude	mi
direction	° north of east of Dallas

Answer 1

Let us consider

d1 = 730 mi in a direction of θ1 = 5.0° north of east

d2 = 560 miles in a direction of θ2 = 21.0° west of north from Atlanta

The net displacement along x-direction is

      dx = d1cosθ1 + d2sinθ2

           = (730 mi)cos5.0° +(560 mile)sin21°

           = 527.0 miles

The net displacement along y-direction is

           dy = d1sinθ1 + d2cosθ2

                =  (730 mi)sin5° +(560 mile)cos21°     

                = 586 mi

The magnitude of the net displacement is

          d = √[dx^2 + dy^2]

             = √[(527mi)^2 + ( 586mi )^2]

             = 788 mi

Direction is

        φ = tan-1[(586 mi )/(527 mi)]

           = 48^o

So the displacement from Dallas to Chicago 788 mi at north east of Dallas