We can solve this problem using vector analysis and trigonometry. The man walks in two legs, and we need to determine his distance from the starting point (which will form a triangle) and his final bearing.

Step 1: Breakdown of the directions

• First, the man walks 5.7 miles in the direction S 26° W. This means the angle is 26 degrees west of the south.
• Then, he turns 90° and walks 9 miles in the direction N W, which means 45 degrees west of north.

We want to determine:

1. The straight-line distance from his final position back to his starting point (i.e., the hypotenuse of a triangle).
2. His bearing from the starting point.

Step 2: Decompose the walks into components

We'll break each leg of his walk into its north-south (vertical) and east-west (horizontal) components using trigonometry.

Leg 1: 5.7 miles at S 26° W

• Southward component: $5.7 \times \cos(26^\circ)$
• Westward component: $5.7 \times \sin(26^\circ)$

Calculating: $\text{Southward component} = 5.7 \times \cos(26^\circ) \approx 5.7 \times 0.8988 = 5.122 \text{ miles south}$ $\text{Westward component} = 5.7 \times \sin(26^\circ) \approx 5.7 \times 0.4384 = 2.500 \text{ miles west}$

Leg 2: 9 miles at N W (45 degrees west of north)

• Northward component: $9 \times \cos(45^\circ)$
• Westward component: $9 \times \sin(45^\circ)$

Since $\cos(45^\circ) = \sin(45^\circ) \approx 0.7071$, we can calculate:

$\text{Northward component} = 9 \times 0.7071 = 6.364 \text{ miles north}$ $\text{Westward component} = 9 \times 0.7071 = 6.364 \text{ miles west}$

Step 3: Total displacement in the north-south and east-west directions

Now we sum the north-south and east-west components.

• North-south direction:

  • He walked 5.122 miles south in the first leg and 6.364 miles north in the second leg.
  • Net displacement: $6.364 - 5.122 = 1.242 \text{ miles north}$

• East-west direction:

  • He walked 2.500 miles west in the first leg and 6.364 miles west in the second leg.
  • Net displacement: $2.500 + 6.364 = 8.864 \text{ miles west}$

Step 4: Calculate the straight-line distance (resultant vector)

The man's straight-line distance from his starting point is the hypotenuse of the right triangle formed by the north-south and east-west displacements. Use the Pythagorean theorem:

$d = \sqrt{(1.242)^2 + (8.864)^2}$

Calculating:

$d = \sqrt{1.542 + 78.573} = \sqrt{80.115} \approx 8.951 \text{ miles}$

Step 5: Determine the bearing

The bearing