To solve this problem, we can treat triangle $ABC$ where the distance $AB$ and the angles $\angle ABC$ and $\angle BAC$ are given. We are tasked with finding the width of the river, which is the perpendicular distance from point $C$ to the line segment $AB$.

We will use the Law of Sines to find the length of side $AC$, which is the distance from point $A$ to point $C$. Once we know $AC$, we can calculate the width of the river using trigonometric properties.

Step-by-Step Solution

1. Assign the triangle components:

   • $AB = 125 \text{ feet}$
   • $\angle ABC = 65^\circ 40'$
   • $\angle BAC = 54^\circ 30'$

2. Find $\angle ACB$: Since the sum of the angles in a triangle is $180^\circ$, we can calculate $\angle ACB$: $\angle ACB = 180^\circ - 65^\circ 40' - 54^\circ 30'$ Converting the angle minutes into degrees: $\angle ACB = 180^\circ - 65^\circ 40' - 54^\circ 30' = 180^\circ - 120^\circ 10' = 59^\circ 50'$

3. Apply the Law of Sines: The Law of Sines is given by: $\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle ABC)} = \frac{BC}{\sin(\angle BAC)}$ We can solve for $AC$ using: $\frac{AC}{\sin(65^\circ 40')} = \frac{125}{\sin(59^\circ 50')}$ Using a calculator to compute the sines: $\sin(65^\circ 40') \approx 0.9096, \quad \sin(59^\circ 50') \approx 0.8634$ Substituting these values into the equation: $\frac{AC}{0.9096} = \frac{125}{0.8634}$ Solving for $AC$: $AC = \frac{125 \times 0.9096}{0.8634} \approx 131.77 \text{ feet}$

4. Calculate the width of the river: The width of the river is the perpendicular distance from $C$ to line $AB$, which can be found using the sine function. Since we now know $AC$ and $\angle BAC$, the width is given by: $\text{Width} = AC \times \sin(\angle BAC)$ Substituting the known values: $\text{Width} = 131.77 \times \sin(54^\circ 30') \approx 131.77 \times 0.8130 \approx 107.14 \text{ feet}$

Thus, the width of the river is approximately 107.14 feet.

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Would you like further details or have any questions about this solution? Here are some related questions to explore:

1. How does the Law of Sines work in non-right triangles?
2. What would happen if the triangle were a right triangle instead?
3. How can you determine the height of a triangle from just two angles and one side?
4. What is the significance of angle measurements in degrees and minutes?
5. How would the result change if the angles were slightly different?

Tip: When working with angles in degrees and minutes, always convert them to decimal form before using them in calculations for better accuracy.