The eye of a hurricane passes over Grand Bahama Island. It is moving in a direction θ = 50° north of west with speed v1 = 34.5 km/h. Exactly three hours later, the course of the hurricane shifts due north, and its speed slows to v2 = 16.1 km/h, as shown below. How far from Grand Bahama is the hurricane 3.5 h after it passes over the island?
To determine the distance of the hurricane from Grand Bahama Island 3.5 hours after it passes over, we need to break down the problem into different components and calculate each separately.
First, let's calculate the distance traveled by the hurricane during the initial three-hour period.
Distance1 = speed1 * time1
Distance1 = 34.5 km/h * 3 hours
Distance1 = 103.5 km
Next, we need to calculate the distance traveled after the course of the hurricane shifts due north for the remaining 0.5 hours.
Distance2 = speed2 * time2
Distance2 = 16.1 km/h * 0.5 hours
Distance2 = 8.05 km
Now, we can calculate the displacement of the hurricane relative to its initial position. We will use vector addition to determine the total displacement.
Displacement = √(Distance1^2 + Distance2^2 + 2 * Distance1 * Distance2 * cosθ)
where θ is the angle between Distance1 and Distance2.
θ = 50° (angle north of west)
Substituting the values into the formula:
Displacement = √((103.5 km)^2 + (8.05 km)^2 + 2 * 103.5 km * 8.05 km * cos 50°)
Calculating the equation:
Displacement = √(10702.25 km^2 + 64.8025 km^2 + 1670.1225 km^2)
Displacement = √(12437.175 km^2)
Displacement ≈ 111.52 km
So, the hurricane is approximately 111.52 km from Grand Bahama Island 3.5 hours after it passes over.