Amap suggests that atlanta is d1 = 729 mi in a direction of θ1 = 5.11° north of east from dallas. the same map shows that chicago is d2 = 562 miles in a direction of θ2 = 21.1° west of north from atlanta. modeling the earth as flat, use this information to find the displacement from dallas to chicago.

We can use the Cosine Law:
c² = a² + b² - 2 a b cos C
d² = 729² + 562² - 2 · 729 · 562 · cos C
5.11 + ( 90 - 21.1 ) = 5.11 + 68.9 = 74.01°
d² = 531,441 + 315,844 + 2 · 729 · 562 · cos 74.01°
d² = 847,285 - 2 · 409,698  · 0.27547 
d² = 847,285 - 225,718.67
d² = 621,566.33
d = √621,566.33
d = 788.4 miles

The displacement from Dallas to Chicago is 788.4 miles.