Final answer:
To find the distance the HMS Sasquatch is from the port, we must solve the vector addition problem graphically by calculating the northward and eastward components for both legs of the journey and using the Pythagorean theorem to find the resultant displacement.
Step-by-step explanation:
The question is asking how far the HMS Sasquatch is from port after changing course. To solve this, we need to apply principles of vector addition and the Pythagorean theorem to add the two vectors representing the ship's movements and determine the resulting displacement from the origin.
The HMS Sasquatch first travels 7 miles on a bearing of N19ºE. This can be represented by a vector pointing 19º east of the north line. It then changes course to S35ºE for 3 miles, which is a vector pointing 35º east of the south line.
To find the distance from the port, we need to break down these movements into their north-south and east-west components and then combine these components to find the resultant displacement vector. The total northward/southward displacement is the sum of the northward component of the first vector minus the southward component of the second vector. The total eastward displacement is the sum of the eastward components of both vectors.
Let's calculate the components:
- First leg northward component = 7 miles × cos(19º)
- First leg eastward component = 7 miles × sin(19º)
- Second leg southward component = 3 miles × cos(35º)
- Second leg eastward component = 3 miles × sin(35º)
Now, the total northward displacement is the first leg's north component minus the second leg's south component, and the total eastward displacement is the sum of the eastward components of both legs. Finally, using the Pythagorean theorem, the distance from port is the square root of the sum of the squares of these northward and eastward displacements.
Calculate the final distance:
- Total northward displacement = 7 × cos(19º) - 3 × cos(35º)
- Total eastward displacement = 7 × sin(19º) + 3 × sin(35º)
- Distance from port = √(Total northward displacement² + Total eastward displacement²)
After evaluating the trigonometric components and summing up the squared displacements, take the square root to find the final distance from port, rounded to the nearest hundredth of a mile.