To find the distance $A B$ across a river, a surveyor laid off a distance $B C=359 \mathrm{~m}$ on one side of the river. It is found that $B=110^{\circ} 30^{\prime}$ and $C=15^{\circ} 10^{\prime}$. Find $A B$.
The distance $A B$ across the river is m (Simplify your answer. Do not round until the final ans

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.