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To find the distance AB across a river, a surveyor laid off a distance BC=363m on one side of the river. It is found that B=111^(@)20^(') and C=13^(@)15^('). Find AB.

To find the distance AB A B across a river, a surveyor laid off a distance BC=363 m B C=363 \mathrm{~m} on one side of the river. It is found that B=111∘20′ \mathrm{B}=111^{\circ} 20^{\prime} and C=13∘15′ \mathrm{C}=13^{\circ} 15^{\prime} . Find AB A B .

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Q. To find the distance AB A B across a river, a surveyor laid off a distance BC=363 m B C=363 \mathrm{~m} on one side of the river. It is found that B=111∘20′ \mathrm{B}=111^{\circ} 20^{\prime} and C=13∘15′ \mathrm{C}=13^{\circ} 15^{\prime} . Find AB A B .
  1. Convert Angles to Decimal Degrees: Convert angles BB and CC from degrees and minutes to decimal degrees for easier calculations. \newlineAngle B=111B = 111 degrees 2020 minutes =111+2060=111.333= 111 + \frac{20}{60} = 111.333 degrees.\newlineAngle C=13C = 13 degrees 1515 minutes =13+1560=13.25= 13 + \frac{15}{60} = 13.25 degrees.
  2. Calculate Angle A: Use the angle sum property of a triangle, where the sum of angles in a triangle is 180180 degrees. Calculate angle A.\newlineAngle A = 180−(Angle B+Angle C)=180−(111.333+13.25)=55.417180 - (\text{Angle B} + \text{Angle C}) = 180 - (111.333 + 13.25) = 55.417 degrees.
  3. Apply Law of Sines for AB: Apply the Law of Sines to find AB: (sin⁡A/AB)=(sin⁡B/BC)(\sin A / AB) = (\sin B / BC). Rearrange to find AB: AB=BC×(sin⁡A/sin⁡B)AB = BC \times (\sin A / \sin B).
  4. Calculate sin⁡A\sin A and sin⁡B\sin B: Calculate sin⁡A\sin A and sin⁡B\sin B using a calculator:\newlinesin⁡A=sin⁡(55.417)≈0.829\sin A = \sin(55.417) \approx 0.829,\newlinesin⁡B=sin⁡(111.333)≈0.932\sin B = \sin(111.333) \approx 0.932.
  5. Substitute Values for AB: Substitute values into the formula: AB=363×(0.829/0.932)≈363×0.889≈322.917AB = 363 \times (0.829 / 0.932) \approx 363 \times 0.889 \approx 322.917 meters.

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