As it passes over Grand Bahama Island, the
eye of a hurricane is moving in a direction 48 degrees north of west with a speed of 13 km/h. Three hours later, it shifts due north, and its speed slows to 75 km/h. How far from Grand Bahama is the eye 4.50h after it passes over the island?
To find the distance of the eye from Grand Bahama 4.50 hours after passing over the island, we need to break down the motion of the eye into its components and calculate the distance traveled in each direction.
First, let's determine the distance traveled during the initial 3 hours when the eye is moving 48 degrees north of west at a speed of 13 km/h.
Step 1: Convert the direction angle to westward and northward components.
To do this, we can use trigonometry. The westward component can be found using the cosine of the angle, and the northward component can be found using the sine of the angle.
Westward component = speed * cos(direction angle)
Northward component = speed * sin(direction angle)
In this case:
Westward component = 13 km/h * cos(48 degrees)
Northward component = 13 km/h * sin(48 degrees)
Step 2: Calculate the total distance traveled during the first 3 hours.
To find the total distance, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Total distance = sqrt((westward component)^2 + (northward component)^2)
Step 3: Calculate the time taken during the first 3 hours.
During the first 3 hours, the eye is moving at a constant speed of 13 km/h.
Time taken = speed * time
= 13 km/h * 3 h
Now, let's determine the distance traveled during the next 1.50 hours when the eye shifts due north and slows down to a speed of 75 km/h.
Step 4: Calculate the distance traveled during the next 1.50 hours.
To find the distance, we multiply the speed by the time.
Distance = speed * time
= 75 km/h * 1.50 h
Step 5: Calculate the total distance traveled after 4.50 hours.
To find the total distance, we add the distance traveled during the first 3 hours to the distance traveled during the next 1.50 hours.
Total distance = distance during the first 3 hours + distance during the next 1.50 hours
Finally, we can calculate the total distance by adding up the distances from the individual steps.
I'll do the calculations for you.
Westward component = 13 km/h * cos(48 degrees) ≈ 8.91 km/h
Northward component = 13 km/h * sin(48 degrees) ≈ 9.85 km/h
Total distance during the first 3 hours = sqrt((8.91 km/h)^2 + (9.85 km/h)^2) ≈ 13.0 km
Time taken during the first 3 hours = 13 km/h * 3 h = 39 km
Distance during the next 1.50 hours = 75 km/h * 1.50 h = 112.5 km
Total distance after 4.50 hours = 13.0 km + 112.5 km = 125.5 km
Therefore, the eye of the hurricane is approximately 125.5 kilometers away from Grand Bahama Island after 4.50 hours.