27. Correct Direction for Crossing a River A river has a constant current of 3 kilometers per hour. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kilometers per hour, be headed in order to reach a point directly opposite the dock? If the river is \( \frac{1}{2} \) kilometer wide, how long will it take to cross?

We want the boat’s net motion to be directly across the river despite the 3‑km/h current. Step 1. Adjusting for the Current Let θ be the angle (relative to directly across the river) at which the boat must be aimed upstream. The boat’s speed relative to the water is 20 km/h. Its component in the horizontal (across the river) direction is 20 cos θ, while its component in the upstream direction is 20 sin θ. The river’s current of 3 km/h pushes the boat downstream. To compensate, the upstream component of the boat’s velocity must cancel the downstream current. Thus, we set   20 sin θ = 3. Solving for θ gives   sin θ = 3/20  →  θ = sin⁻¹(3/20). Numerically, sin⁻¹(0.15) is approximately 8.6°. So the boat must be aimed about 8.6° upstream from the perpendicular (i.e. from the line directly across the river). Step 2. Finding the Crossing Time The effective speed directly across the river is given by the component of the boat’s velocity perpendicular to the bank,   20 cos θ. Using the fact that cos θ = √(1 – sin²θ), we have   cos θ = √(1 – (3/20)²) = √(1 – 9/400) = √(391/400) = √391/20. Thus, the effective speed is   20 cos θ = 20 (√391/20) = √391 km/h. However, it is easier to directly compute the time: The river is ½ km wide, so   time = (distance)/ (speed across)      = (½ km) / (20 cos θ). Substitute cos θ:   time = 0.5 / (20 × (√391/20)) = 0.5 / √391 hours. Calculating numerically, note that √391 is approximately 19.8, so   time ≈ 0.5/19.8 ≈ 0.0253 hours. To convert to minutes, multiply by 60:   0.0253 × 60 ≈ 1.52 minutes. Summary of Answers: • The motorboat should be headed at an angle θ where sin θ = 3/20, i.e. approximately 8.6° upstream from the straight‐across direction. • The crossing will take about 1.5 minutes.