two boats leave the same port at the same time. one travels at a speed of 29 mph in the direction N 50 degrees E and the other travels at a speed of 35 mph in the direction S 70 degrees E. how far apart are the two boats after an hour.
To determine the distance between the two boats after an hour, we need to calculate the positions each boat will be at after one hour of travel.
Let's start with Boat A, which is traveling at a speed of 29 mph in the direction N 50 degrees E.
We can break down the velocity into its x and y components using trigonometry. The x-component (horizontal) can be given by cos(theta) * speed, and the y-component (vertical) can be given by sin(theta) * speed.
For Boat A:
x-component = cos(50 degrees) * 29 mph
y-component = sin(50 degrees) * 29 mph
Next, we need to calculate how far Boat A will travel in one hour. We'll assume a constant speed.
For Boat A:
distance = speed * time
distance = 29 mph * 1 hour
Now, we can calculate the final position of Boat A. Since it's moving in the N 50 degrees E direction, the x-component will be added to the starting x position, and the y-component will be added to the starting y position.
For Boat A:
final x position = starting x position + x-component * time
final y position = starting y position + y-component * time
Now let's do the same calculations for Boat B, which is traveling at a speed of 35 mph in the direction S 70 degrees E.
For Boat B:
x-component = cos(70 degrees) * 35 mph
y-component = sin(70 degrees) * 35 mph
For Boat B:
distance = 35 mph * 1 hour
For Boat B:
final x position = starting x position - x-component * time
final y position = starting y position + y-component * time
Finally, we can calculate the distance between the final positions of the two boats using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the final positions of the two boats into the formula will give us the distance between them after one hour.
To find the distance between the two boats after an hour, we can use the concept of vector addition.
First, let's calculate the displacement of each boat after an hour.
Boat 1: The boat is traveling at a speed of 29 mph in the direction N 50 degrees E. This can be represented as a vector with a magnitude of 29 and a direction of 50 degrees from the north direction.
To calculate the displacement, we can use the formula:
displacement = speed × time
displacement of Boat 1 = 29 mph × 1 hour = 29 miles
Boat 2: The boat is traveling at a speed of 35 mph in the direction S 70 degrees E. This can be represented as a vector with a magnitude of 35 and a direction of 70 degrees from the south direction.
To calculate the displacement, we can use the formula:
displacement = speed × time
displacement of Boat 2 = 35 mph × 1 hour = 35 miles
Now, let's represent the displacements of both boats as vectors and add them using vector addition.
To add two vectors, we can break them down into their x and y components, add the corresponding components, and then calculate the magnitude and direction of the resulting vector.
Displacement of Boat 1 (Vector A):
Magnitude = 29 miles
Direction = 50 degrees from the north direction
Displacement of Boat 2 (Vector B):
Magnitude = 35 miles
Direction = 70 degrees from the south direction
To find the x and y components of each vector, we can use trigonometric functions.
Vector A:
x component = magnitude of A × cos(direction of A)
y component = magnitude of A × sin(direction of A)
Vector B:
x component = magnitude of B × cos(direction of B)
y component = magnitude of B × sin(direction of B)
Now, let's calculate the x and y components for both vectors:
Vector A:
x component = 29 miles × cos(50 degrees)
y component = 29 miles × sin(50 degrees)
Vector B:
x component = 35 miles × cos(70 degrees)
y component = 35 miles × sin(70 degrees)
After calculating the x and y components of both vectors, we can add the corresponding components:
Total x component = x component of A + x component of B
Total y component = y component of A + y component of B
Now, to find the magnitude and direction of the resulting vector, we can use the formula:
Resultant magnitude = √(total x component^2 + total y component^2)
Resultant direction = atan2(total y component, total x component)
By substituting the values, we can find the magnitude and direction of the resulting vector.
Finally, the distance between the two boats after an hour would be the magnitude of the resulting vector.