Ray and Roxanne leave from the same spot for their morning run. Roxanne runs due North at 8 km/h while Ray runs due East at 11 km/h. Find the distance and bearing of Roxanne from Ray after 20 minutes. Round your answers to one decimal place.
To find the distance and bearing of Roxanne from Ray after 20 minutes, we can break down the problem into two components: the distance traveled by each person and the resulting displacement between them.
First, let's calculate the distance traveled by each person:
Roxanne's distance = speed × time
Roxanne's distance = 8 km/h × (20 minutes / 60 minutes)
Roxanne's distance = 8 km/h × (1/3) hours
Roxanne's distance = (8/3) km
Ray's distance = speed × time
Ray's distance = 11 km/h × (20 minutes / 60 minutes)
Ray's distance = 11 km/h × (1/3) hours
Ray's distance = (11/3) km
Now, let's calculate the displacement between Ray and Roxanne using the Pythagorean theorem:
Displacement = √((Roxanne's distance)^2 + (Ray's distance)^2)
Displacement = √((8/3)^2 + (11/3)^2)
Displacement = √((64/9) + (121/9))
Displacement = √(185/9)
Displacement ≈ 4.055 km (rounded to one decimal place)
To find the bearing, we will use the inverse tangent function:
Bearing = arctan(Ray's distance / Roxanne's distance)
Bearing = arctan((11/3) / (8/3))
Bearing ≈ 54.5 degrees (rounded to one decimal place)
Therefore, after 20 minutes, Roxanne is approximately 4.1 km away from Ray, with a bearing of 54.5 degrees.