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.