The points P, Q and B lie in a horizontal plane. From P, which is due west of B, the angle of elevation of the top of a tower AB of height h metres is 37 degrees. From Q, which is on a bearing of 213 degrees from the tower, the angle of elevation of the top of the tower is 41 degrees. The distance PQ is 225 metres

To solve this problem, let’s define the positions and use trigonometry to determine relevant distances and relationships.

### Step 1: Understanding Positions and Angles
- Let \( A \) be the top of the tower, and \( B \) be the base of the tower.
- \( P \) is located due west of \( B \), meaning the angle \( \angle PBA \) is aligned horizontally.
- From \( P \), the angle of elevation to \( A \) is \( 37^\circ \).
- \( Q \) is positioned on a bearing of \( 213^\circ \) from \( B \) (or \( BQ \) forms an angle of \( 213^\circ \) from North).
- The angle of elevation from \( Q \) to \( A \) is \( 41^\circ \).
- The distance \( PQ = 225 \) meters.

### Step 2: Setting Up Coordinate Geometry
To set up coordinates, we can place point \( B \) at the origin \((0,0)\):
1. Since \( P \) is due west of \( B \), let \( P = (-d, 0) \).
2. Let the height of the tower be \( h \), so \( A = (0, h) \).

### Step 3: Using the Angle of Elevation from \( P \)
The angle of elevation from \( P \) to \( A \) is \( 37^\circ \), so we can use the tangent function:
\[
\tan(37^\circ) = \frac{h}{d}
\]
Solving for \( h \):
\[
h = d \cdot \tan(37^\circ)
\]

### Step 4: Using the Position of \( Q \)
To place \( Q \) accurately using the bearing of \( 213^\circ \):
- The angle \( 213^\circ \) corresponds to \( 213 - 180 = 33^\circ \) south-west from \( B \), or an angle \( 33^\circ \) with the south-west direction.
- Let \( Q = (x, y) \) in the coordinate system centered at \( B \), and we can later find its coordinates with respect to \( P \).

Using trigonometry and further analysis of both angles, we could now find both the height \( h \) and distances

The points P, Q and B lie in a horizontal plane. From P, which is due west of B, the angle of elevation of the top of a tower AB of height h metres is 37 degrees. From Q, which is on a bearing of 213 degrees from the tower, the angle of elevation of the top of the tower is 41 degrees. The distance PQ is 225 metres.

This solution explores trigonometric methods to find the height of a tower from given angles of elevation at two points and bearings. Using right triangle trigonometry, we analyze positions, angles, and coordinates in a horizontal plane, ideal for high school students in Grades 10-12.

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