A wildlife researcher is tracking a flock of geese. the geese fly 3.5 km due west, then turn toward the north by 40 ∘ and fly another 4.5 km, what is the magnitude of their displacement

To solve this problem, we can use the cosine formula for calculating the length of the displacement:

c^2 = a^2 + b^2 - 2 a b cos θ

where c is the displacement, a = 3.5 km, b = 4.5 km, and θ is the angle inside the triangle

Since the geeze turned 40° from west to north, so the angle inside the triangle must be:

θ = 180 - 40 = 140°

c^2 = 3.5^2 + 4.5^2 - 2 (3.5) (4.5) cos 140

c^2 = 56.63

c = 7.53 km

So the magnitude of the displacement is 7.53 km