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To solve this problem, we can use trigonometry, specifically the tangent function, which relates the opposite side and the adjacent side of a right triangle.

### Problem Analysis:
- The biologist walks 100 feet downstream from point A to the new sighting point.
- The angle \( \theta = 58^\circ \).
- The width of the river is represented by \( w \), which is the opposite side of the angle.

### Trigonometric Formula:
The tangent of the angle is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting the given values:

\[
\tan(58^\circ) = \frac{w}{100}
\]

### Solving for \( w \):

\[
w = 100 \times \tan(58^\circ)
\]

Let's calculate this value.The width of the river is approximately \( 160.0 \) feet.

Would you like further details or have any questions?

### Related Questions:
1. How would the river's width change if the angle was different?
2. What if the biologist walked a different distance downstream?
3. How can this method be used in real-life scenarios beyond measuring river width?
4. What are the limitations of using trigonometry in such measurements?
5. How can errors in angle measurement affect the calculation?

**Tip:** Always ensure the angle measurement is accurate, as small errors can significantly affect the final result.

A biologist wants to know the width (w) of a river. The biologist walks downstream 100 feet from point A and sights to point C. From this sighting, it is determined that the angle θ = 58°. How wide is the river?

Learn how to calculate the width of a river using trigonometry. A biologist walks 100 feet downstream and sights an angle of 58° to determine the river's width. This problem covers the tangent function and is suitable for students in Grades 9-12.

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