Math Snap

STEP 1

1. The problem involves a triangle $\triangle ABC$ where $B$ and $C$ are angles, and $BC$ is a known side.
2. The angle $B$ is $115^\circ 30'$.
3. The angle $C$ is $13^\circ 15'$.
4. The side $BC$ is $351$ meters.
5. We need to find the distance $AB$.

STEP 2

1. Calculate the third angle $A$ of the triangle.
2. Use the Law of Sines to find the length of $AB$.

STEP 3

Calculate the third angle $A$ of the triangle using the angle sum property of triangles:
$$A = 180^\circ - B - C$$ Substitute the given angles:
$$A = 180^\circ - 115^\circ 30' - 13^\circ 15'$$ Convert the angles to decimal form for easier calculation:
$$115^\circ 30' = 115.5^\circ, \quad 13^\circ 15' = 13.25^\circ$$ Calculate:
$$A = 180^\circ - 115.5^\circ - 13.25^\circ = 51.25^\circ$$

Use the Law of Sines to find $AB$:
$$\frac{AB}{\sin C} = \frac{BC}{\sin A}$$ Substitute the known values:
$$\frac{AB}{\sin 13.25^\circ} = \frac{351}{\sin 51.25^\circ}$$ Solve for $AB$:
$$AB = \frac{351 \times \sin 13.25^\circ}{\sin 51.25^\circ}$$ Calculate the sines:
$$\sin 13.25^\circ \approx 0.2298, \quad \sin 51.25^\circ \approx 0.7771$$ Substitute these values:
$$AB = \frac{351 \times 0.2298}{0.7771} \approx 103.8$$ Round to the nearest whole number:
$$AB \approx 104 \text{ meters}$$ The distance $AB$ across the river is:
$$\boxed{104 \text{ meters}}$$