Step 1 :Given a triangle ABC, where BC = 359 m, angle B = 110°30' and angle C = 15°10'. We are asked to find the length of side AB.
Step 2 :First, we convert the given angles from degrees to radians. We find that B = 1.928588823453734 radians and C = 0.26470826988580665 radians.
Step 3 :We can use the law of sines to find the length of side AB. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Step 4 :Applying the law of sines, we get \(\frac{AB}{\sin B} = \frac{BC}{\sin C}\). Solving for AB, we get \(AB = BC \cdot \frac{\sin B}{\sin C}\).
Step 5 :Substituting the given values, we find that AB = 1285.2817285491733 m.
Step 6 :Rounding to two decimal places, we get the final answer: The distance AB across the river is \(\boxed{1285.28}\) m.