Vector addition of boat and river current velocities

Steps:

• Boat's velocity components: perpendicular to banks = 6 cos θ m/s; upstream parallel = 6 sin θ m/s.
• For resultant velocity perpendicular to banks, net parallel component = 0: 6 sin θ - 4 = 0 → sin θ = 4/6 = 2/3 → θ = arcsin(2/3) ≈ 42°.
• cos θ = √(1 - sin² θ) = √(1 - 4/9) = √(5/9) = √5 / 3 ≈ 0.745.
• Resultant speed v = perpendicular component = 6 cos θ = 6 × (√5 / 3) = 2√5 ≈ 4.5 m/s.

Why A is correct:

• Matches exact calculation: θ ≈ 42° from sin θ = current/boat speed ratio; v = 4.5 m/s from Pythagorean theorem on components.

Why the others are wrong:

• B: v = 7.2 m/s exceeds boat's maximum speed (6 m/s), ignoring vector cancellation.
• C: θ = 48° from incorrect sin θ = 6/4 (swapped speeds), yielding wrong angle.
• D: Combines C's wrong θ with B's excessive v.

Final answer: A