A map 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

Answer:

The displacement from Dallas to Chicago is 788.4 miles.