A light house P was 40 km and N32degreesW from Q. A ship moved in the direction of S60degreesW, leaving from P. When The ship was nearest to Q, it changed the course to N20degreesW. The ship reached R which is north of Q

To solve this problem, we need to determine the position of the ship at point R, and we will do this by breaking the journey into segments and analyzing each part of the path.

### Step 1: Plot the positions
1. **Lighthouse P:** Given as 40 km from point Q, at a direction of N32°W from Q. We can place Q at the origin (0, 0) of a coordinate system, and then use trigonometry to find the coordinates of P.
   - Distance from Q to P = 40 km.
   - Bearing of N32°W translates to an angle of 32° west of north.
   - The coordinates of P can be found using:
     \[
     P_x = -40 \sin(32^\circ)
     \]
     \[
     P_y = 40 \cos(32^\circ)
     \]

2. **Calculating \(P_x\) and \(P_y\):**
   \[
   P_x = -40 \sin(32^\circ) \approx -21.2 \text{ km}
   \]
   \[
   P_y = 40 \cos(32^\circ) \approx 33.9 \text{ km}
   \]
   Thus, the coordinates of P are approximately (-21.2, 33.9).

### Step 2: Ship's journey
3. **First leg from P in the direction of S60°W:**
   - This direction means 60° west of south, or equivalently, 240° on a standard coordinate system.
   - We need to determine the path when the ship is closest to Q. This will occur when the ship's path is perpendicular to the line connecting Q and P.
   - The slope of line QP can be calculated as:
     \[
     \text{slope of QP} = \frac{33.9 - 0}{-21.2 - 0} = -\frac{33.9}{21.2} \approx -1.6
     \]
   - The perpendicular slope will be the negative reciprocal, which is approximately \(0.625\).
   - We can find the point where the ship is nearest to Q by solving the system of equations representing the ship’s path and the perpendicular slope.

### Step 3: Change direction at the nearest point
4. **New course of N20°W:**
   - This is 20° west of north.
   - We will follow this direction to reach point R, ensuring it is due north of Q (same x-coordinate as Q).

### Step 4: Determine the coordinates of R
5. **Calculate the final leg to find R's position** based on the directional change and making sure the x-coordinate aligns with Q.

This outline involves setting up equations based on the given directions, distances, and slopes. Solving these equations step by step gives the exact coordinates of R.

Would you like a more detailed step-by-step calculation for each leg? 

### Related Questions
1. How do you determine the coordinates of a point given a distance and bearing?
2. What is the method for finding the closest point on a path to a fixed location?
3. How can you convert between directional bearings and standard coordinate angles?
4. What mathematical principles are used to solve navigation problems in geometry?
5. How does one solve for unknowns in a system of linear equations derived from slope and intercept forms?

### Tip
For navigation problems, always start by converting the bearings to a standard coordinate angle and plot the known points to establish a reference frame.

A light house P was 40 km and N32degreesW from Q. A ship moved in the direction of S60degreesW, leaving from P. When the ship was nearest to Q, it changed the course to N20degreesW. The ship reached R which is north of Q.

Learn how to solve a navigational problem involving a lighthouse and a ship changing its course to find the nearest point to a given location. This solution covers key trigonometric concepts and coordinate geometry, suitable for students in Grades 10-12.

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