As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 25.0° north of east with a speed of 40.6 km/h. Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows to 23.0 km/h. How far from Grand Bahama is the eye 7.40 h after the hurricane turned to the north?
call north y and east x
phase one:
vx1 = 40.6 cos 25
vy1 = 40.6 sin 25
x1 = 40.6 cos 25 * 3
y1 = 40.6 sin 25 * 3
Phase 2
vx2 = 0
vy2 = 23
x2 = x1 + 0 * 7.4
y2 = y1 + 23 * 7.4
d = sqrt (x2^2 + y2^2)
To solve this problem, we can break it down into two parts: the initial movement of the hurricane and its subsequent movement after changing course.
Part 1:
The initial movement of the hurricane is described as moving in a direction 25.0° north of east with a speed of 40.6 km/h.
To determine the distance covered by the hurricane during the initial 3 hours, we can use the formula:
Distance = Speed * Time
Distance (Part 1) = 40.6 km/h * 3 hours
Distance (Part 1) = 121.8 km
Now, let's move on to Part 2:
Part 2:
After 3 hours, the course of the hurricane suddenly shifts due north, and its speed slows to 23.0 km/h.
To determine the distance covered by the hurricane during this time, we can use the formula:
Distance = Speed * Time
Distance (Part 2) = 23.0 km/h * 4.40 h (7.40 h - 3 h)
Distance (Part 2) = 101.2 km
To find the total distance from Grand Bahama Island after 7.40 hours, we can add the distances covered during Part 1 and Part 2:
Total Distance = Distance (Part 1) + Distance (Part 2)
Total Distance = 121.8 km + 101.2 km
Total Distance = 223 km
Therefore, the eye of the hurricane is approximately 223 km from Grand Bahama Island 7.40 hours after the hurricane turned to the north.
To find the distance from Grand Bahama Island to the eye of the hurricane 7.40 hours after it turned north, we need to break down the problem into different components and calculate them step by step.
First, we need to find the distance the eye of the hurricane traveled in the initial direction (25.0° north of east) for the first 3 hours. We can do this by using the formula:
Distance traveled = Speed * Time
Distance1 = 40.6 km/h * 3 hours
Next, we need to find the distance the eye of the hurricane traveled in the new north direction for the remaining time. We can calculate this using:
Distance2 = Speed * Time
Distance2 = 23.0 km/h * (7.40 hours - 3 hours)
Now, we need to find the total distance from Grand Bahama Island to the eye of the hurricane. This can be found by calculating the vector sum of Distance1 and Distance2 using the Pythagorean theorem:
Total distance = √(Distance1^2 + Distance2^2)
Note: Since we are dealing with vectors, we need to consider the components separately.
Horizontal component = Distance1 + Distance2 * cos(90° - 25.0°)
Vertical component = Distance2 * sin(90° - 25.0°)
Finally, we can find the total distance using:
Total distance = √(Horizontal component^2 + Vertical component^2)
By substituting the earlier calculated values, we can find the answer.