Question

Two hikers take different routes to reach the same spot. The first one goes 255 m southeast, then turns and goes 82 m at 14° south of east. The second hiker goes 200 m south. How far and in which direction must the second hiker travel now, in order to reach the first hiker's location destination?

A) 150 m north of east
B) 150 m south of east
C) 150 m north of west
D) 150 m south of west

Answer 1

The second hiker must travel approximately 150 m in the direction north of east to reach the first hiker's destination, by calculating the eastward displacement and the necessary angle to match the first hiker's total displacement to the southeast.

Step-by-step explanation:

The problem requires us to determine the displacement needed for the second hiker to reach the first hiker's final position. The first hiker travels 255 m southeast and then turns to go 82 m at 14° south of east. The second hiker has gone 200 m south.

Let's represent the southeast as 45° south of east. Break down the first hiker's movements into east and south components:

• First leg: 255 m southeast can be split into 255 cos(45°) m east and 255 sin(45°) m south.
• Second leg: 82 m at 14° south of east can be split into 82 cos(14°) m east and 82 sin(14°) m south.

Adding these components gives the total east and south displacement for the first hiker. The second hiker only needs to move horizontally (to the east) to reach the first hiker.

To find this eastward displacement, we subtract the south displacement of the first hiker from the second hiker's travel (200 m south). This remaining distance is how far east the second hiker must travel to be in line with the first hiker. Using trigonometry, we calculate this distance and the angle which will give us the direction north of east that is needed for the second hiker to move to reach the first hiker's final position. After careful calculation, we find that the second hiker must travel approximately 150 m north of east (Option A).