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 the island, we'll break the problem down into two parts:
1. First, we'll calculate the distance the hurricane traveled in the three-hour period after passing over Grand Bahama Island.
2. Then, we'll calculate the distance the hurricane traveled in the remaining 0.5 hours after the course shifted due north.
Let's begin with the first part:
1. Distance traveled in the three-hour period:
We have the speed of the hurricane during this period, v1 = 34.5 km/h, and the time, t1 = 3 hours. To calculate the distance traveled, we'll use the formula:
Distance (d1) = Speed (v1) x Time (t1)
Plugging in the values:
d1 = 34.5 km/h x 3 hours
d1 = 103.5 km
Therefore, the hurricane traveled a distance of 103.5 km in the three-hour period.
Now, let's move on to the second part:
2. Distance traveled in the remaining 0.5 hours:
After the course shifted due north, the speed of the hurricane is given as v2 = 16.1 km/h, and the time is t2 = 0.5 hours.
Similarly, using the formula:
Distance (d2) = Speed (v2) x Time (t2)
Plugging in the values:
d2 = 16.1 km/h x 0.5 hours
d2 = 8.05 km
Therefore, the hurricane traveled a distance of 8.05 km in the remaining 0.5 hours.
To find the total distance from Grand Bahama Island 3.5 hours after the hurricane passed over, we add the distance traveled in the first part to the distance traveled in the second part:
Total distance = d1 + d2
Total distance = 103.5 km + 8.05 km
Total distance ≈ 111.55 km
Therefore, the hurricane is approximately 111.55 km away from Grand Bahama Island 3.5 hours after passing over the island.