What is the distance between city A and city C?

Question

An airplane flies 233 km due west from city A to city B and then 440 km in the direction of 36◦ north of west from city B to city C. What is the distance between city A and city C? Answer in units of km.

Answer 1

To find the distance between city A and city C, we can use the concept of vector addition. We'll break down the displacement into its horizontal and vertical components.

Given:

Distance from city A to city B (AB) = 233 km (due west)

Distance from city B to city C (BC) = 440 km (36° north of west)

First, let's find the horizontal and vertical components of the displacement BC:

Horizontal component (BCx) = BC * cos(36°)

Vertical component (BCy) = BC * sin(36°)

BCx = 440 km * cos(36°)

BCy = 440 km * sin(36°)

Next, we can find the net horizontal and vertical displacements by adding the corresponding components:

Net horizontal displacement = AB + BCx

Net vertical displacement = BCy

Net horizontal displacement = 233 km + 440 km * cos(36°)

Net vertical displacement = 440 km * sin(36°)

Now, we can use the Pythagorean theorem to find the magnitude of the displacement between city A and city C:

Distance between city A and city C = √(Net horizontal displacement)^2 + (Net vertical displacement)^2

Distance between city A and city C = √[(233 km + 440 km * cos(36°))^2 + (440 km * sin(36°))^2]