A rural mail carrier leaves the post office and drives 22.0 km in a direction 35 degrees to the next town. She then drives in a direction 60.0 degrees for 47 km. What is her magnitude and displacement from the post office?

D = 72.96 km,      θ = 36.43º

Explanation:

This is a vector addition exercise, we are going to solve it by decomposing the displacements.

First displacement

         cos 35 = x₁ / L

         x₁ = L cos 35

         cos 35 = y₁ / L

         y₁ = L sin 35

         x₁ = 22.0 cos 35 = 18.0 km

         y₁ = 22 sin 35 = 12,619 km

Second displacement

         cos 60 = x₂ / L

         x₂ = L cos 60

         x₂ = 47 cos 60 = 23.5 km

        sin 60 = y₂ / L

        y₂ = L sin 60

        y₂ = 47 sin 60 = 40.70 m

In total displacement it is

       x_total = x₁ + x₂

       x_total = 18 + 40.7 = 58.7 km

       y_total = y₁ + y₂

       y_total = 12.629 +40.70

       y_total = 53,329 km

Let's use the Pythagorean theorem

        D = √(x_total² + y_total² )

        D = √ (58.7² + 43,329²)

        D = 72.96 km

For the direction we use trigonometry

         tan θ = y_total / x_total

         θ= tan-1 (43,329 / 58.7)

         θ = 36.43º

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