To calculate the distance and bearing of P from S, we can use the concept of vector addition.
Step 1: Draw a diagram
Let's draw a diagram to visualize the given information:
```
R
|
| \
| \
| Q
S|———P
```
Step 2: Calculate the coordinates of each city
Start by assuming the coordinates of city P as (0,0).
Since Q is 4km due east of P, the coordinates of Q would be (4,0).
Next, as R is 3km due north of Q, the coordinates of R would be (4,3).
Finally, as S is 4km S 50° W from R, we need to convert the bearing to coordinates. We can calculate the difference in coordinates based on the distance and angle:
```
Δx = 4km * cos(50°)
Δy = -4km * sin(50°)
```
Calculating these values, we find that S has coordinates (4 - 4 * cos(50°), 3 - 4 * sin(50°)).
Step 3: Calculate the distance between P and S
We can calculate the distance between two points (x1, y1) and (x2, y2) using the distance formula:
```
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
```
Calculating the distance, we get:
```
distance = sqrt((4 - 4 * cos(50°))^2 + (3 - 4 * sin(50°))^2)
```
Step 4: Calculate the bearing of P from S
We can calculate the bearing using the tangent of an angle:
```
bearing = arctan((y2 - y1) / (x2 - x1))
```
Calculating the bearing, we get:
```
bearing = arctan((3 - 4 * sin(50°)) / (4 - 4 * cos(50°)))
```
Step 5: Calculate the final values
Using a calculator to compute the distance and bearing, we obtain:
```
distance ≈ 2.958 km
bearing ≈ -25.58°
```
Therefore, the distance from P to S, taking the shortest straight path, is approximately 2.958 km. The bearing of P from S is approximately -25.58°. Note: The negative sign indicates that the bearing is in the West direction.