To solve the problem, we'll break it down step by step.

Step 1: Understand the Bearing Information

• The bearing of a point is the direction measured in degrees clockwise from the north.
• The bearing of point A from point P is $310^\circ$.
• The bearing of point B from point P is $292^\circ$.

Since bearings are measured clockwise from the north:

• $310^\circ$ means A is slightly west of north.
• $292^\circ$ means B is even further west of north.

Step 2: Visualize and Set Up the Triangle

We are given that both points A and B are due north of a point C, which is 5600 meters due west of P. This setup implies that the distance between A and B is simply the horizontal (west-east) distance between them, as both points lie on the same vertical line (due north).

Given that bearings are measured clockwise from the north:

• The angle $\theta_1$ from the north to PA (from P to A) is $360^\circ - 310^\circ = 50^\circ$.
• The angle $\theta_2$ from the north to PB (from P to B) is $360^\circ - 292^\circ = 68^\circ$.

Step 3: Calculate the Horizontal Distances to Points A and B

Using trigonometry, we can calculate the horizontal distances (westward components) to points A and B.

For point A: $x_A = \text{Distance from P to C} \times \tan(\theta_1) = 5600 \times \tan(50^\circ)$

For point B: $x_B = \text{Distance from P to C} \times \tan(\theta_2) = 5600 \times \tan(68^\circ)$

Step 4: Calculate the Distance Between A and B

The distance between A and B, $d_{AB}$, is the difference between the horizontal distances to A and B: $d_{AB} = |x_B - x_A|$

Let’s calculate it.

$x_A = 5600 \times \tan(50^\circ) \approx 5600 \times 1.1918 \approx 6674.08 \text{ meters}$ $x_B = 5600 \times \tan(68^\circ) \approx 5600 \times 2.4751 \approx 13860.56 \text{ meters}$

Thus, the distance between A and B is: $d_{AB} = |13860.56 - 6674.08| \approx 7186.48 \text{ meters}$

Final Answer

The distance between points A and B is approximately 7186.48 meters.

Would you like further details on any part of the solution or have any questions?

Follow-up Questions

1. How would the solution change if point C was east of P instead of west?
2. What would the bearing of A and B be if they were southeast of P?
3. How would you calculate the distance between A and B if they were not due north of point C?
4. What trigonometric principles are involved in solving this problem?
5. How would the problem change if P was at a different location, say, due south of C?

Tip

When dealing with bearings, always convert them to angles measured from the north in the counterclockwise direction to simplify calculations.