Width of river from bearings

Let T be the tree on the opposite bank and PQ the base line of 400 m on the near bank. The width of the river is the perpendicular distance of T from the line PQ.

Step 1: Bearings in whole circle form

         T (tree, far bank)
        / \
       /   \
  ----P-----Q---- near bank
        400 m

Step 2: Angles of the triangle PQT

$$\angle TPQ = 94^\circ 30' - 40^\circ 30' = 54^\circ 00'$$ $$\angle TQP = 322^\circ 15' - 274^\circ 30' = 47^\circ 45'$$ $$\angle PTQ = 180^\circ - 54^\circ 00' - 47^\circ 45' = 78^\circ 15'$$

Step 3: Sine rule

$$PT = \frac{PQ \sin \angle TQP}{\sin \angle PTQ} = \frac{400 \times \sin 47^\circ 45'}{\sin 78^\circ 15'} = 302.42\ \text{m}$$

Step 4: Width of river

$$\text{Width} = PT \sin \angle TPQ = 302.42 \times \sin 54^\circ = 244.67\ \text{m}$$

(Check with the single formula: $\dfrac{400 \sin 54^\circ \sin 47^\circ 45'}{\sin 78^\circ 15'} = 244.67$ m.)

Answer: width of the river = 244.67 m ≈ 244.7 m.