A man wandering in the desert walks 5.7 miles in the direction S 26 degrees﻿﻿ W. He then turns ﻿﻿90 degrees and walks 9 miles in the direction N 49 degrees﻿﻿ W. At that time, how far is he from his starting point, and what is his bearing from his starting point?

To find the man's distance from the starting point and his bearing, we can model his path as a vector problem. Here's a step-by-step solution:

### Step 1: Represent the displacements as vectors.
We first break down each leg of his journey into horizontal (x) and vertical (y) components using trigonometry.

#### First leg: 5.7 miles in the direction \( S \ 26^\circ W \)
This means he is walking 26° west of due south.
- The horizontal component (west direction) is:
  \[
  x_1 = 5.7 \times \sin(26^\circ)
  \]
- The vertical component (south direction) is:
  \[
  y_1 = 5.7 \times \cos(26^\circ)
  \]
  Since this is towards the south, we treat \( y_1 \) as negative.

#### Second leg: 9 miles in the direction \( N \ 49^\circ W \)
This means he is walking 49° west of due north.
- The horizontal component (west direction) is:
  \[
  x_2 = 9 \times \sin(49^\circ)
  \]
- The vertical component (north direction) is:
  \[
  y_2 = 9 \times \cos(49^\circ)
  \]
  Since this is towards the north, we treat \( y_2 \) as positive.

### Step 2: Calculate the total displacement.
Now, add up the horizontal and vertical components to get the total displacement.
- Total horizontal displacement:
  \[
  x_{\text{total}} = x_1 + x_2
  \]
- Total vertical displacement:
  \[
  y_{\text{total}} = -y_1 + y_2
  \]

### Step 3: Find the distance from the starting point.
The total distance is the magnitude of the resultant vector, given by:
\[
d = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}
\]

### Step 4: Find the bearing.
The bearing is the angle \( \theta \) measured clockwise from the north direction, calculated using:
\[
\theta = \tan^{-1} \left( \frac{|x_{\text{total}}|}{|y_{\text{total}}|} \right)
\]
Once you have \( \theta \), adjust it to find the correct bearing based on the quadrant of the resultant vector.

Let's now compute these values.The man is approximately 9.32 miles from his starting point. His bearing from the starting point is about \( 85.19^\circ \) west of north (or approximately \( N 85.19^\circ W \)).

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:
1. How do you convert from Cartesian coordinates to polar coordinates?
2. What is the significance of the angle direction in bearings?
3. How would the path change if the man walked due west instead of turning?
4. How can vector addition be used to solve more complex navigation problems?
5. How would the solution change if we used kilometers instead of miles?

**Tip:** Always make sure to check the quadrant when calculating angles to get the correct bearing.

A man wandering in the desert walks 5.7 miles in the direction S 26 degrees W. He then turns 90 degrees and walks 9 miles in the direction N 49 degrees W. At that time, how far is he from his starting point, and what is his bearing from his starting point?

Solve a navigation problem where a man walks 5.7 miles S 26° W, then 9 miles N 49° W. Learn how to calculate the total distance and bearing using trigonometry, vector addition, and the Pythagorean theorem. Suitable for students in Grades 10-12.

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