To determine the width of the river, let's analyze the given data step-by-step. Here's the setup based on the information provided:

1. Known Distances and Angles:

   • The distance between poles $A$ and $B$ is $100 \, \text{m}$.
   • The angle of elevation to the top of pole $A$ from point $M$ is $22^\circ$.
   • The angle of elevation to the top of pole $B$ from point $M$ is $19^\circ$.
   • Let $D$ represent the width of the river (i.e., the horizontal distance from $M$ to $A$ or $B$, since the poles are directly across the river from $M$).

2. Using Right Triangles: Let:

   • $h$ represent the height of the poles $A$ and $B$.
   • We can set up two right triangles, one with angle $22^\circ$ for pole $A$ and one with angle $19^\circ$ for pole $B$.

3. Setting Up Tangent Equations:

   • For pole $A$: $\tan(22^\circ) = \frac{h}{D}$ So, $h = D \cdot \tan(22^\circ)$

   • For pole $B$: $\tan(19^\circ) = \frac{h}{D + 100}$ So, $h = (D + 100) \cdot \tan(19^\circ)$

4. Equating the Two Expressions for $h$: Since both expressions represent $h$, we can set them equal to each other: $D \cdot \tan(22^\circ) = (D + 100) \cdot \tan(19^\circ)$

5. Solving for $D$: Expanding and rearranging terms: $D \cdot \tan(22^\circ) = D \cdot \tan(19^\circ) + 100 \cdot \tan(19^\circ)$ $D \cdot (\tan(22^\circ) - \tan(19^\circ)) = 100 \cdot \tan(19^\circ)$ $D = \frac{100 \cdot \tan(19^\circ)}{\tan(22^\circ) - \tan(19^\circ)}$

6. Calculating the Value of $D$: Using approximate values for the tangents:

   • $\tan(22^\circ) \approx 0.4040$
   • $\tan(19^\circ) \approx 0.3443$

   Substitute these values: $D = \frac{100 \cdot 0.3443}{0.4040 - 0.3443}$ $D \approx \frac{34.43}{0.0597}$ $D \approx 576.57 \, \text{m}$

Therefore, the width of the river is approximately $576.57 \, \text{m}$.

Would you like more details on any of these steps or further clarification?

Related Questions

1. How can we verify this calculation using a different trigonometric approach?
2. What assumptions are made in this solution, and how would errors in angle measurement affect the result?
3. How would the calculation change if the poles were of different heights?
4. Can we use sine or cosine rules to solve similar problems in different configurations?
5. How could you use similar principles to measure the height of a building using two points on the ground?

Tip

When solving trigonometric problems, drawing a clear diagram helps in visualizing relationships and setting up equations correctly.