Final answer:
The helicopter is 6√3 miles away from the landing pad after flying 6 miles east and then 6 miles southwest. It must fly in a direction of 60° to return directly to the landing pad, which means option a is correct.
Step-by-step explanation:
The student's question asks about the distance and direction a helicopter is from its landing pad after flying a specific path. To solve this, we need to use vector addition and trigonometry. First, the helicopter flies 6 miles east, which can be represented as a vector pointing directly to the right on a grid. Then, it flies 6 miles southwest. The direction southwest is 45° south of west or 225° from the east direction on the compass. Now, we need to determine the resultant vector by breaking down the southwest travel into south and west components. Since it forms a 45° angle, the components in the south and west directions have the same magnitude. For a 6-mile distance at a 45° angle, each component will be 6/√2 miles.
To find the resultant displacement from the starting point, we subtract the west component from the east travel because they are in opposite directions, while the south component remains as it is, given there is no other south/north travel. Mathematically, the east-west displacement is 6 - 6/√2 miles and the north-south displacement is 6/√2 miles. To find the total displacement, we use the Pythagorean theorem: s = √[(6 - 6/√2)² + (6/√2)²]. This simplifies to s = 6√3 miles. To find the direction, θ, back to the starting point, we take the arctan of the south component over the east-west component, which gives us θ = arctan(6/√2 / (6 - 6/√2)). This simplifies to θ = 60°.
Therefore, the helicopter is 6√3 miles away from the landing pad and must fly in a direction of 60° to head directly back to the landing pad. Hence, the correct answer to the student's question is option a) (s = 6√3) miles, (θ = 60°).