To solve this problem, we can use trigonometry and vector addition. Let's break it down step by step, starting with the diagram:
1. Draw a diagram with a point representing the port. Label it as "P."
2. From point P, draw a line segment in the direction of bearing 135°. Label the endpoint of this line as "A," representing the first boat after 2 hours.
3. From point P, draw another line segment in the direction of bearing 063°. Label the endpoint of this line as "B," representing the second boat after 2 hours.
Now, let's calculate their distance apart:
1. Determine the distance traveled by the first boat after 2 hours:
Distance = Speed × Time
Distance = 15 km/hr × 2 hr = 30 km
So, the first boat is 30 km away from the port.
2. Determine the distance traveled by the second boat after 2 hours:
Distance = Speed × Time
Distance = 20 km/hr × 2 hr = 40 km
So, the second boat is 40 km away from the port.
3. Now, we need to find the actual distance between the two boats.
To do this, we can use vector addition.
4. Draw a line segment connecting points A and B on your diagram.
This line represents the vector from the first boat to the second boat.
5. We can break down this vector into its x and y components.
The x-component represents the east-west direction, and the y-component represents the north-south direction.
For the first boat (A):
- x-component (A): Distance × cos(bearing 135°)
= 30 km × cos(135°)
= -21.2 km (negative because it is west of the port)
- y-component (A): Distance × sin(bearing 135°)
= 30 km × sin(135°)
= 21.2 km (positive because it is north of the port)
For the second boat (B):
- x-component (B): Distance × cos(bearing 063°)
= 40 km × cos(63°)
= 20 km
- y-component (B): Distance × sin(bearing 063°)
= 40 km × sin(63°)
= 35.5 km
6. Add the x-components and y-components separately to find the resulting vector.
- x-component (resultant): (x-component A) + (x-component B)
= -21.2 km + 20 km
= -1.2 km (west direction)
- y-component (resultant): (y-component A) + (y-component B)
= 21.2 km + 35.5 km
= 56.7 km (north direction)
7. Calculate the magnitude (distance) of the resulting vector using the Pythagorean theorem:
Distance = sqrt((x-component)^2 + (y-component)^2)
Distance = sqrt((-1.2 km)^2 + (56.7 km)^2)
= sqrt(1.44 km^2 + 3214.89 km^2)
= sqrt(3216.33 km^2)
= 56.7 km
So, the distance between the two boats after 2 hours is approximately 56.7 km.