Oasis B is 8.0 km due east of oasis A. Starting from oasis A, a camel walks20 km in a direction 15.0° south of east and then
walks32 km due north. If it is to then walk directly to B, (a) how far and (b) in what direction (relative to the positive x-axis within the range (-180°, 180°]) should it walk?
To find the distance and direction from Oasis A to Oasis B, we can break down the camel's movements into its horizontal and vertical components.
First, let's find the camel's net displacement in the x (horizontal) direction.
Given that the camel walks 20 km in a direction 15.0° south of east, we can calculate the horizontal component of its displacement:
Horizontal displacement = 20 km * cos(15.0°)
Next, let's find the camel's net displacement in the y (vertical) direction.
Given that the camel walks 32 km due north, we know that the vertical displacement is simply 32 km.
Now, let's calculate the total displacement from Oasis A to Oasis B by adding the horizontal and vertical components.
Total displacement = √[(Horizontal displacement)² + (Vertical displacement)²]
To find the direction, we can use the inverse tangent function:
Direction = atan(Vertical displacement / Horizontal displacement)
Let's plug in the given values and calculate the distance and direction:
Horizontal displacement = 20 km * cos(15.0°) ≈ 19.422 km
Vertical displacement = 32 km
Total displacement = √[(19.422 km)² + (32 km)²] ≈ 37.464 km
Direction = atan(32 km / 19.422 km) ≈ 58.014°
Therefore, the camel should walk approximately 37.464 km in a direction of 58.014° (relative to the positive x-axis) to go directly from Oasis A to Oasis B.