Final answer:
The HMS Sasquatch's distance from port is calculated using vector addition and the Pythagorean theorem, resolving each leg of its journey into north/south and east/west components and then finding the resultant vector.
Step-by-step explanation:
The HMS Sasquatch leaves port on a bearing of N19°E and travels for 7 miles, then changes course to S35°E for an additional 3 miles. To calculate the distance from port, we need to use vector addition and the Pythagorean theorem. First, each leg of the trip can be seen as a vector, with the first leg being 7 miles in a north and east combination, and the second leg being 3 miles in a south and east combination. We need to resolve these into their north/south and east/west components.
For the first leg, the northward component is 7 cos(19°) and the eastward component is 7 sin(19°). For the second leg, which is in the south-east direction, we will have a southward component of 3 sin(35°) and an eastward component of 3 cos(35°).
However, since south is the opposite direction from north, we must treat the south component as negative when adding them. Hence, the northward displacement is 7 cos(19°) - 3 sin(35°) and the eastward displacement is 7 sin(19°) + 3 cos(35°). To find the distance from port, which is the resultant vector, we square each component, sum them, and take the square root of that sum — this is the application of the Pythagorean theorem — and we obtain √[(7 cos(19°) - 3 sin(35°))^2 + (7 sin(19°) + 3 cos(35°))^2].