Suppose you would like to cross a 200-foot wide river in a boat. Assume that the boat can travel 35 mph relative to the water and that the current is flowing west at the rate of 5 mph. What bearing should be chosen so that the boat will land at a point exactly across from its starting point? Give your answer to the nearest tenth of a degree.
If it takes x hours to cross the river as intended, and
200ft = .06096 km
0.06096^2 + (5x)^2 = (35x)^2
x = 0.00175976
tanθ = 5*0.00175976 / 0.06096 = 0.14433
θ = 8.2°
hmm. or, with less calculation,
sinθ = 5/35 = 1/7
θ = 8.2°
To determine the bearing needed for the boat to land directly across from its starting point, we can break down the problem into two components: the boat's velocity (in still water) and the current's velocity.
First, let's determine the boat's velocity relative to the ground. Since the boat can travel 35 mph relative to the water, and the current is flowing west at 5 mph, we can subtract the current speed from the boat's speed to find the boat's velocity relative to the ground:
Boat's Velocity relative to the ground = Boat's velocity relative to the water - Current's velocity
Boat's Velocity relative to the ground = 35 mph - 5 mph
Boat's Velocity relative to the ground = 30 mph
Next, let's calculate the time it would take for the boat to cross the river.
Time = Distance / Velocity
Since the river width is given as 200 feet, and the boat's velocity relative to the ground is 30 mph, we need to convert the units to be consistent.
Converting 200 feet to miles, we divide by 5,280 (the number of feet in a mile):
200 feet / 5,280 feet/mile = 0.03788 miles
Now, we can calculate the time it would take for the boat to cross the river:
Time = 0.03788 miles / 30 mph
Time = 0.00126 hours
Now that we know the time it takes for the boat to cross the river, we can calculate how far downstream the boat will drift due to the current during that time.
Distance Drifted downstream = Current's velocity x Time
Distance Drifted downstream = 5 mph x 0.00126 hours
Distance Drifted downstream = 0.0063 miles
To determine the bearing that will take the boat directly across from its starting point, we can use trigonometry. We have a right-angled triangle where the boat's velocity relative to the ground (30 mph) represents the hypotenuse, and the distance drifted downstream (0.0063 miles) represents the side adjacent to the angle we want to find.
Tan (θ) = Opposite / Adjacent
Tan (θ) = Distance Drifted downstream / Boat's Velocity relative to the ground
Tan (θ) = 0.0063 miles / 30 mph
Now, let's solve for θ:
θ = arctan (0.0063 miles / 30 mph)
Using a calculator, we find:
θ ≈ 0.0126 degrees
Rounded to the nearest tenth of a degree, the bearing that should be chosen is approximately 0.0 degrees.