To determine the width of the river (distance $d$ from point $M$ to point $A$ on the opposite bank), we can use trigonometry, as the problem provides two angles of elevation and the distance between poles $A$ and $B$ (100 m).

Let's break down the solution step-by-step:

1. Define the Variables and Set Up Triangles:

   • Let $h$ represent the height of both poles $A$ and $B$.
   • $d$ is the width of the river (the horizontal distance from $M$ to $A$).
   • The angle of elevation from $M$ to the top of pole $A$ is $22^\circ$.
   • The angle of elevation from $M$ to the top of pole $B$ is $19^\circ$.
   • The distance between $A$ and $B$ is 100 m.

2. Use the Tangent Function: For the two angles of elevation, we can set up the following equations based on the tangent of each angle:

   $\tan(22^\circ) = \frac{h}{d}$ $\tan(19^\circ) = \frac{h}{d + 100}$

3. Solve for $h$ in Terms of $d$: From the first equation: $h = d \cdot \tan(22^\circ)$

   Substitute this expression for $h$ in the second equation: $d \cdot \tan(22^\circ) = (d + 100) \cdot \tan(19^\circ)$

4. Expand and Solve for $d$: Distribute $\tan(19^\circ)$: $d \cdot \tan(22^\circ) = d \cdot \tan(19^\circ) + 100 \cdot \tan(19^\circ)$

   Rearrange to isolate $d$: $d (\tan(22^\circ) - \tan(19^\circ)) = 100 \cdot \tan(19^\circ)$

   Now, solve for $d$: $d = \frac{100 \cdot \tan(19^\circ)}{\tan(22^\circ) - \tan(19^\circ)}$

5. Calculate $d$: Substitute the values for the tangents (using approximate values for simplicity):

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

   $d = \frac{100 \cdot 0.3443}{0.4040 - 0.3443}$

   $d \approx \frac{34.43}{0.0597}$

   $d \approx 576.8 \text{ meters}$

Conclusion

The width of the river $d$ is approximately 576.8 meters.

Would you like more details on any part of this calculation or have further questions?

Related Questions:

1. How would the result change if the angles were closer together?
2. Could we solve this problem with other trigonometric functions, like sine or cosine?
3. How does the distance between poles $A$ and $B$ affect the result?
4. What would happen if the poles were of different heights?
5. How accurate would the measurement be if angles were rounded?

Tip:

Always check for alternate approaches using trigonometric identities for complex angle-based problems.