A sailboat heads north at 3 m/s for 1 hour and then tacks back to the southwest (at 45 degree to north) at 2 m/s for 45 minutes.

a. How far has the boat sailed?
b. How far is it from its starting location?

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To calculate the distance the boat has sailed, we need to first calculate the distance traveled when moving north and then when moving southwest.

a. Distance traveled when moving north:
The speed of the boat when moving north is 3 m/s, and it travels in this direction for 1 hour. Therefore, the distance traveled when moving north can be calculated using the formula: distance = speed × time.

Distance when moving north = 3 m/s × 1 hour
Distance when moving north = 3 m/s × 3600 seconds (since 1 hour = 3600 seconds)
Distance when moving north = 10,800 meters

b. Distance traveled when moving southwest:
When the boat tacks back to the southwest (at a 45-degree angle to north), we can break down the movement into two components: one in the north direction and the other in the west direction.

The speed of the boat when moving southwest is 2 m/s. The boat travels at this speed for 45 minutes, but it's more convenient to convert this time to seconds: 45 minutes = 45 × 60 seconds = 2700 seconds.

The distance traveled in the northward direction can be calculated similarly to part (a), considering that the angle of the boat's movement is 45 degrees to north. We need to calculate the component of the speed in the north direction first: northward speed = speed × cos(angle). Substituting the values, we get:

northward speed = 2 m/s × cos(45 degrees)
northward speed ≈ 2 m/s × 0.707 (taking the cosine of 45 degrees)
northward speed ≈ 1.414 m/s

Distance traveled in the northward direction = northward speed × time
Distance traveled in the northward direction = 1.414 m/s × 2700 seconds
Distance traveled in the northward direction ≈ 3814.2 meters

The distance traveled in the westward direction can be calculated similarly to the northward direction, but this time using the sine of the angle (since it's the component perpendicular to the north direction):

westward speed = speed × sin(angle)
westward speed = 2 m/s × sin(45 degrees)
westward speed ≈ 2 m/s × 0.707 (taking the sine of 45 degrees)
westward speed ≈ 1.414 m/s

Distance traveled in the westward direction = westward speed × time
Distance traveled in the westward direction = 1.414 m/s × 2700 seconds
Distance traveled in the westward direction ≈ 3814.2 meters

To find the total distance traveled when moving southwest, we can use the Pythagorean theorem to find the hypotenuse of the right triangle formed by the north and west components of the distance traveled:

Total distance traveled when moving southwest = sqrt((distance north)^2 + (distance west)^2)
Total distance traveled when moving southwest = sqrt((3814.2 meters)^2 + (3814.2 meters)^2)
Total distance traveled when moving southwest ≈ 5397.4 meters

c. Total distance sailed:
To find the total distance sailed, we add up the distance traveled when moving north and the distance traveled when moving southwest:

Total distance sailed = Distance when moving north + Total distance traveled when moving southwest
Total distance sailed = 10,800 meters + 5397.4 meters
Total distance sailed ≈ 16,197.4 meters

d. Distance from starting location:
To find the distance from the starting location, we calculate the straight line distance from the starting location to the final location. This is the hypotenuse of the right triangle formed by the distance traveled when moving north and the distance traveled when moving southwest.

Distance from starting location = sqrt((Distance when moving north)^2 + (Total distance traveled when moving southwest)^2)
Distance from starting location = sqrt((10,800 meters)^2 + (5397.4 meters)^2)
Distance from starting location ≈ 12,240 meters

So, the sailboat has sailed approximately 16,197.4 meters total, and it is approximately 12,240 meters from its starting location.