Math  /  Trigonometry

QuestionTo find the distance ABA B across a river, a surveyor laid off a distance BC=351 mB C=351 \mathrm{~m} on one side of the river. It is found that B=11530\mathrm{B}=115^{\circ} 30^{\prime} and C=1315\mathrm{C}=13^{\circ} 15^{\prime}. Find ABA B.
The distance ABA B across the river is \square m (Simplify your answer. Do not round until the final answer. Then round to the nearest whole number as needed.)

Studdy Solution
Use the Law of Sines to find AB AB :
ABsinC=BCsinA\frac{AB}{\sin C} = \frac{BC}{\sin A}
Substitute the known values:
ABsin13.25=351sin51.25\frac{AB}{\sin 13.25^\circ} = \frac{351}{\sin 51.25^\circ}
Solve for AB AB :
AB=351×sin13.25sin51.25AB = \frac{351 \times \sin 13.25^\circ}{\sin 51.25^\circ}
Calculate the sines:
sin13.250.2298,sin51.250.7771\sin 13.25^\circ \approx 0.2298, \quad \sin 51.25^\circ \approx 0.7771
Substitute these values:
AB=351×0.22980.7771103.8AB = \frac{351 \times 0.2298}{0.7771} \approx 103.8
Round to the nearest whole number:
AB104 metersAB \approx 104 \text{ meters}
The distance AB AB across the river is:
104 meters \boxed{104 \text{ meters}}

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