As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.7° north of west with a speed of 37.4km/hr. Two hours later, the course of the hurricane suddenly shifts due north and its speed slows to 21.0km/hr. How far from Grand Bahama is the eye 5.35hr after it passes over the island?
you have velocities, times, bearings.
so you can get distances and bearings. Add those two vectors.
To solve this problem, we can break it down into two parts: the first two hours when the hurricane is moving 60.7° north of west, and the latter 5.35 hours when the hurricane shifts due north.
First, let's calculate the distance the eye of the hurricane travels in the first two hours.
Step 1: Calculate the horizontal distance (x-coordinate) traveled in the first two hours.
To find the horizontal distance, we use the formula:
Horizontal distance = Speed × Time × Cos(θ)
Given:
Speed = 37.4 km/hr
Time = 2 hr
θ (angle north of west) = 60.7°
Calculating the horizontal distance:
Horizontal distance = 37.4 km/hr × 2 hr × Cos(60.7°)
Using a calculator, we find:
Horizontal distance = 37.4 km/hr × 2 hr × 0.5009
Step 2: Calculate the vertical distance (y-coordinate) traveled in the first two hours.
To find the vertical distance, we use the formula:
Vertical distance = Speed × Time × Sin(θ)
Given:
Speed = 37.4 km/hr
Time = 2 hr
θ (angle north of west) = 60.7°
Calculating the vertical distance:
Vertical distance = 37.4 km/hr × 2 hr × Sin(60.7°)
Using a calculator, we find:
Vertical distance = 37.4 km/hr × 2 hr × 0.8590
Now, let's calculate the distance the eye of the hurricane travels in the latter 5.35 hours.
Step 3: Calculate the horizontal distance traveled in the latter 5.35 hours.
To find the horizontal distance, we use the formula:
Horizontal distance = Speed × Time
Given:
Speed = 21.0 km/hr
Time = 5.35 hr
Calculating the horizontal distance:
Horizontal distance = 21.0 km/hr × 5.35 hr
Using a calculator, we find:
Horizontal distance = 21.0 km/hr × 5.35 hr
Finally, to find the total distance from Grand Bahama Island, we need to calculate the resultant distance using the horizontal and vertical distances.
Step 4: Calculate the resultant distance.
To find the resultant distance, we use the Pythagorean theorem:
Resultant distance = √(Horizontal distance^2 + Vertical distance^2)
Calculating the resultant distance:
Resultant distance = √((Horizontal distance first 2 hours + Horizontal distance latter 5.35 hours)^2 + (Vertical distance first 2 hours)^2)
Using a calculator, we find:
Resultant distance = √((Horizontal distance first 2 hours)^2 + (Vertical distance first 2 hours)^2 + (Horizontal distance latter 5.35 hours)^2)
Now, you can substitute the values we calculated in Steps 1, 2, 3, and 4 to find the final answer.