Suppose you would like to cross a 204 ft wide river in a boat. Assume that the boat can travel 37 mph relative to the water and the current is flowing westbat a rate of 6mph. What bearing should be chosen so that the boat will land at a point exactly across from the starting point?

Aim upstream at angle

A = sin^-1(6/37) = 9.3 degrees

The upstream component of the boat's velocity relative to the water will then cancel out the river's flow speed.

To determine the bearing the boat should choose, we can break down the velocities involved and use simple trigonometry.

Let's start with the boat's speed relative to the ground.
The boat's speed relative to the water is 37 mph, and the current is flowing westwards at a rate of 6 mph. Therefore, the boat's speed relative to the ground is the vector sum of these two velocities. We can use the Pythagorean theorem to find this resultant velocity:

Ground Speed = √(Boat Speed^2 + Current Speed^2)
Ground Speed = √(37^2 + 6^2)
Ground Speed ≈ √(1414 + 36)
Ground Speed ≈ √1450
Ground Speed ≈ 38.08 mph (approximately)

Now that we know the boat's ground speed, we need to determine the time it takes to cross the river. We can use the formula:

Time = Distance / Speed

Distance = 204 ft (width of the river)
Speed = Ground Speed = 38.08 mph

Converting 204 ft to miles (since speed is in mph):
204 ft ≈ 0.0386 miles (approximately)

Time = 0.0386 miles / 38.08 mph
Time ≈ 0.001013 hours (approximately)

Next, let's find the heading or bearing the boat should take. We can use trigonometry to determine the angle.

Tangent(θ) = Opposite / Adjacent
Tangent(θ) = Current Speed / Boat Speed
Tangent(θ) = 6 mph / 37 mph
Tangent(θ) ≈ 0.1622

Now, we can find the angle θ by taking the arctangent of 0.1622:

θ ≈ arctan(0.1622)
θ ≈ 9.236 degrees (approximately)

Therefore, the boat should choose a bearing of approximately 9.236 degrees so that it will land at a point exactly across from the starting point.