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To solve for the width of the river (distance \( d \) from point \( M \) to line \( AB \)), we can apply trigonometric principles based on the angles of elevation given and the known distance between poles \( A \) and \( B \).

Here's a step-by-step outline of the solution:

1. **Define Heights and Angles**:
   - Let the height of both poles \( A \) and \( B \) be \( h \).
   - The angle of elevation to pole \( A \) from \( M \) is \( 22^\circ \).
   - The angle of elevation to pole \( B \) from \( M \) is \( 19^\circ \).
   - The distance between poles \( A \) and \( B \) is \( 100 \) meters.

2. **Trigonometric Relationships**:
   - From point \( M \) to point \( A \), the angle of elevation is \( 22^\circ \), so we can write:
     \[
     \tan(22^\circ) = \frac{h}{d}
     \]
   - From point \( M \) to point \( B \), the angle of elevation is \( 19^\circ \), and since \( A \) and \( B \) are \( 100 \) meters apart, we can write:
     \[
     \tan(19^\circ) = \frac{h}{d + 100}
     \]

3. **Set Up Equations**:
   - We now have two equations:
     \[
     h = d \cdot \tan(22^\circ)
     \]
     \[
     h = (d + 100) \cdot \tan(19^\circ)
     \]

4. **Equate and Solve for \( d \)**:
   - By equating the expressions for \( h \):
     \[
     d \cdot \tan(22^\circ) = (d + 100) \cdot \tan(19^\circ)
     \]
   - Expand and rearrange to isolate \( d \):
     \[
     d \cdot (\tan(22^\circ) - \tan(19^\circ)) = 100 \cdot \tan(19^\circ)
     \]
     \[
     d = \frac{100 \cdot \tan(19^\circ)}{\tan(22^\circ) - \tan(19^\circ)}
     \]

5. **Calculate \( d \)**:
   - Using approximate values for \( \tan(22^\circ) \approx 0.4040 \) and \( \tan(19^\circ) \approx 0.3443 \):
     \[
     d = \frac{100 \cdot 0.3443}{0.4040 - 0.3443} \approx \frac{34.43}{0.0597} \approx 576.6 \, \text{meters}
     \]

Therefore, the width of the river is approximately \( 576.6 \) meters.

Would you like more details or have any questions?

Here are five related questions:
1. How would the solution change if the angles of elevation were different?
2. Can this method be applied if poles \( A \) and \( B \) were at different heights?
3. How would we approach the problem if the distance between poles \( A \) and \( B \) were unknown?
4. What if we only had one angle of elevation and the height of the pole?
5. How could this method be used in real-world applications beyond river width measurement?

**Tip:** When solving for distances in trigonometry, always ensure angles are accurate, as small errors can lead to large miscalculations in results.

A man, M, positions himself on a river bank so he can observe two poles A and B of equal height on the opposite bank. The angle of elevation to the top of pole A is 22°, and the angle of elevation to the top of pole B is 19°. Given that A and B are 100 m apart, determine the width of the river.

Learn how to calculate the width of a river by using trigonometric ratios and angles of elevation. This problem explores key trigonometric concepts, including the tangent ratio and right triangles, suitable for Grade 10 students.

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