The distance and bearing of town D from school.

Here are the solutions to the questions:

a) i) The distance and bearing of town D from school.

Step 1: Establish a coordinate system with the school (S) at the origin $(0, 0)$.

• School (S): $(0, 0)$

Step 2: Determine the coordinates of Town A.
Town A is 160 km North of the school.

• Town A: $(0, 160)$

Step 3: Determine the coordinates of Town B.
Town B is 100 km Westwards from Town A.

• Town B: $(0 - 100, 160) = (-100, 160)$

Step 4: Determine the coordinates of Town C.
From Town B, the chairperson headed S75°W for 90 km. This means 75° West of South.
The change in x-coordinate ($\Delta x$) is negative (West), and the change in y-coordinate ($\Delta y$) is negative (South).

$$\Delta x = -90 \sin(75^\circ)$$ $$\Delta y = -90 \cos(75^\circ)$$

Using approximate values for $\sin(75^\circ) \approx 0.9659$ and $\cos(75^\circ) \approx 0.2588$:

$$\Delta x = -90 \times 0.9659 = -86.931 \text{ km}$$ $$$$