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.