Question

A boat leaves port and follows a course of N72°E at 6 knots for 3 hr and 20 min. Then, the boat changes to a new course of S27°E at 12 knots for 5 hr. How far is the boat from port?

Answer 1

To find the distance of the boat from port, we need to break down the motion into its horizontal and vertical components and calculate the total displacement using vector addition.

Step-by-step explanation:

To solve this problem, we need to break down the motion of the boat into its horizontal and vertical components. First, we will convert the time taken for each leg of the journey into hours: 3 hours and 20 minutes is equal to 3.33 hours, and 5 hours is equal to 5 hours.

For the first leg of the journey, traveling at a speed of 6 knots in a direction of N72°E for 3.33 hours:

Horizontal Component = 6 knots * cos(72°) * 3.33 hours

Vertical Component = 6 knots * sin(72°) * 3.33 hours

For the second leg of the journey, traveling at a speed of 12 knots in a direction of S27°E for 5 hours:

Horizontal Component = 12 knots * cos(27°) * 5 hours

Vertical Component = -12 knots * sin(27°) * 5 hours (opposite direction)

Once we have the horizontal and vertical components for each leg, we can add them together to find the total displacement of the boat from its starting point. Finally, we can use the Pythagorean theorem to calculate the distance from the starting point.

Answer 2

The boat is approximately 63.23 nautical miles away from the port.

The Breakdown

To find the distance of the boat from the port, we can break down the boat's journey into two segments and calculate the distance traveled in each segment.

Segment 1: N72°E at 6 knots for 3 hours and 20 minutes.

To calculate the distance traveled in this segment, we can use the formula:

Distance = Speed × Time

Converting 3 hours and 20 minutes to hours:

3 hours + (20 minutes / 60 minutes per hour) = 3.33 hours

Distance in segment 1 = 6 knots × 3.33 hours = 19.98 nautical miles

Segment 2: S27°E at 12 knots for 5 hours.

Distance in segment 2 = 12 knots × 5 hours = 60 nautical miles

Now, to find the total distance from the port, we can use the Pythagorean theorem since the two segments form a right triangle.

Total distance = √(Distance segment 1² + Distance segment 2²)

Total distance = √(19.98² + 60²)

Total distance ≈ √(399.2 + 3600)

Total distance ≈ √(3999.2)

Total distance ≈ 63.23 nautical miles

Therefore, the boat is 63.23 nautical miles away from the port.