In the figure given, a radar screen shows 2 ships. Ship A is located at a distance of 40 nautical miles and bearing of 155* , and ship B is located at a distance of 60 nautical miles and bearing 325*
. Which of the
following is an expression for the straight-line distance (dashed line), in nautical miles, between the 2 ships? (Note: for any triangle ABC with side length of a opposite LA, and side length of b opposite LB,
and side length of c opposite LC, the law of cosines states c' = a? + 62 - 2ab cos C.)

To determine the straight-line distance between the two ships, we will use the law of cosines. The information provided includes the distances of the ships from a common point (the radar screen) and their respective bearings. Here's a step-by-step approach:

1. **Understand the Problem Setup:**
   - Ship A is at a distance of \( 40 \) nautical miles and a bearing of \( 155^\circ \).
   - Ship B is at a distance of \( 60 \) nautical miles and a bearing of \( 325^\circ \).

2. **Calculate the Angle Between the Bearings:**
   - The bearings are measured clockwise from the north direction.
   - Bearing of Ship A: \( 155^\circ \)
   - Bearing of Ship B: \( 325^\circ \)

   To find the angle \( C \) between these bearings:
   \[
   \text{Angle between bearings} = 325^\circ - 155^\circ = 170^\circ
   \]

3. **Apply the Law of Cosines:**
   - According to the law of cosines, for any triangle with sides \( a \), \( b \), and \( c \), and the angle opposite to side \( c \) being \( C \), the formula is:
     \[
     c^2 = a^2 + b^2 - 2ab \cos(C)
     \]
   - Here, \( a \) and \( b \) are the distances of the ships from the radar screen, and \( C \) is the angle between their bearings.
   - Let \( a = 40 \) nautical miles, \( b = 60 \) nautical miles, and \( C = 170^\circ \).

4. **Calculate the Distance \( d \):**
   \[
   d^2 = 40^2 + 60^2 - 2 \cdot 40 \cdot 60 \cdot \cos(170^\circ)
   \]
   \[
   d^2 = 1600 + 3600 - 2 \cdot 40 \cdot 60 \cdot \cos(170^\circ)
   \]
   \[
   d^2 = 5200 - 4800 \cdot \cos(170^\circ)
   \]

5. **Simplify the Expression:**
   The final expression for the straight-line distance \( d \) between the two ships is:
   \[
   d = \sqrt{5200 - 4800 \cdot \cos(170^\circ)}
   \]

This expression represents the straight-line distance between the two ships in nautical miles.

Would you like more details or have any questions? 

Here are some related questions you might find useful:

1. How do you convert bearings to standard angles in trigonometry?
2. What is the law of sines, and how does it differ from the law of cosines?
3. How would the calculation change if one of the ships was at a different bearing?
4. Can you explain how to find the cosine of an angle using a calculator?
5. How do you interpret nautical miles in terms of standard units like kilometers or miles?
6. What are some practical applications of the law of cosines in navigation?
7. Can you solve a similar problem with different distances and bearings?
8. How would you visualize this problem using a coordinate system?

**Tip:** When dealing with trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) corresponding to the angle measurements you're using.

Learn how to calculate the straight-line distance between two ships using the law of cosines. This problem involves trigonometric calculations based on ship distances and bearings. Suitable for Grades 10-12.

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