Question

The HMS Sasquatch leaves port on a bearing of N27°E and travels for 9 miles. It then changes course and follows a heading of S45°E for 4 miles. How far is it from port? Round your answer to the nearest hundredth of a mile. What is its bearing to port? Round your angle to the nearest degree. Distance miles Bearing with angle rounded to the nearest degree is ?

Answer 1

To determine the HMS Sasquatch's distance from port and its bearing back to port, vector displacements were calculated using trigonometry and added. The Pythagorean theorem was used to calculate the straight-line distance, and arctan function was used to find the adjusted bearing.

Step-by-step explanation:

To find the distance of the HMS Sasquatch from port after its journey and its bearing to port, we need to analyze the ship's movements as two vector displacements and apply the principles of vector addition and trigonometry.

The first displacement is 9 miles on a bearing of N27°E, which can be visualized on a coordinate system with the positive Y-axis pointing North and the positive X-axis pointing East. The second displacement is 4 miles on a bearing of S45°E.

First displacement (9 miles N27°E):
• Northward component = 9 * cos(27°) miles
• Eastward component = 9 * sin(27°) miles

Second displacement (4 miles S45°E):
• Southward component = 4 * cos(45°) miles
• Eastward component = 4 * sin(45°) miles

Now, we subtract the southward component from the northward component to find the total northward displacement and add the two eastward components to find the total eastward displacement.

Total northward displacement = northward component of first displacement - southward component of second displacement.
• Total eastward displacement = eastward component of first displacement + eastward component of second displacement.

Then, we use the Pythagorean theorem to find the straight-line distance from port:
• Distance from port = √(total northward displacement² + total eastward displacement²).

To find the bearing from the final position back to port, we use the arctan function:
• Bearing to port = arctan(total northward displacement / total eastward displacement). We adjust the bearing according to the compass directions, ensuring it is expressed as an angle in the usual navigational format.

Answer 2

The distance from port is approximately 8.91 miles, and the bearing to port is N52°E.

Explanation:

The solution involves vector addition to determine the displacement from the starting point. First, we use the given bearings and distances to find the horizontal and vertical components of the ship's displacement.

The initial bearing of N27°E translates to a horizontal component of 9 * cos(27°) ≈ 7.98 miles, and a vertical component of 9 * sin(27°) ≈ 4.06 miles. Similarly, the second leg, with a bearing of S45°E, contributes a horizontal component of 4 * cos(45°) ≈ 2.83 miles and a vertical component of 4 * sin(45°) ≈ 2.83 miles. Adding these components gives us the final displacement vector.

To find the magnitude of the displacement (distance from port), we use the Pythagorean theorem: √((7.98 + 2.83)² + (4.06 + 2.83)²) ≈ 8.91 miles. The bearing to port is determined by finding the angle formed by the horizontal and vertical components of the displacement vector.

Using trigonometry, we find the arctangent of the ratio of the vertical to horizontal components: arctan((4.06 + 2.83) / (7.98 + 2.83)) ≈ 52°. Therefore, the ship is approximately 8.91 miles from port at a bearing of N52°E. This method accurately calculates the ship's position based on the provided information.