The eye of a hurricane passes over the Grand Bahama Island in the direction of 60° north of west with the speed of 39.0 km/. three hours later the course of the hurricane suddenly shifts due north and its speed slows to 22.0 km/h. How far from the Grand Bahama is the hurricane 4.70H after it passes over the hou how far from the Grand Bahama is the hurricane 4.70 h after it passes over the island? Please explain in detail struggling to figure this one out.
North position at end = (39 sin 60) (3) + (22) (1.7)
West position at end = (39 cos 60) (3)
for distance square those and take square root of sum of squares
To determine the distance of the hurricane from the Grand Bahama Island 4.70 hours after it passes over the island, we can apply the principles of vectors and trigonometry.
First, let's break down the problem into two parts: the initial movement of the hurricane and its subsequent change in direction and speed.
1. Initial Movement:
The hurricane moves in the direction of 60° north of west with a speed of 39.0 km/h for three hours. To find the total displacement during this time, we can calculate the vector addition.
Since the direction is given as 60° north of west, we need to split it into its westward and northward components. Using trigonometric functions, we find:
- Westward component: 39.0 km/h * cos(60°) = 39.0 km/h * 0.5 = 19.5 km/h
- Northward component: 39.0 km/h * sin(60°) = 39.0 km/h * 0.866 = 33.774 km/h
Next, we multiply the westward component by the time (three hours) to find the westward displacement:
Westward displacement = 19.5 km/h * 3 h = 58.5 km
Similarly, we multiply the northward component by the time to find the northward displacement:
Northward displacement = 33.774 km/h * 3 h = 101.322 km
So, the initial displacement is the vector sum of the westward and northward displacements:
Initial displacement = sqrt((58.5 km)^2 + (101.322 km)^2) ≈ 116.969 km
2. Change in Direction and Speed:
After three hours, the course of the hurricane suddenly shifts due north, and its speed slows to 22.0 km/h. To determine the displacement during the subsequent time, we multiply the new speed by the time (4.70 hours) to find the total displacement.
The displacement in the north direction is given by:
Northward displacement = 22.0 km/h * 4.70 h = 103.4 km
3. Total Displacement:
To find the total displacement, we use the Pythagorean theorem to calculate the magnitude of the final displacement vector:
Total displacement = sqrt((116.969 km)^2 + (103.4 km)^2) ≈ 152.241 km
Therefore, the hurricane is approximately 152.241 km from the Grand Bahama Island 4.70 hours after passing over the island.