Question

Carrie sets off on her hike from the lodge at a bearing of s80ºw. after 1.5 miles, she changes her bearing to s19ºw and continues hiking for 3 miles. find carrie's distance from the lodge at this point.

Answer 1

To determine Carrie's distance from the lodge, we first calculate the components of her two displacements using cosine and sine functions, then apply vector addition and the Pythagorean theorem to find the magnitude and direction of her resultant displacement.

Step-by-step explanation:

The student is asking about a problem related to vectors and displacement, specifically involving bearings and distances. To solve for Carrie's distance from the lodge after changing her direction and walking further, we need to apply the concept of vector addition and use trigonometry to find the resultant vector's magnitude and direction.

First, we find the north and west components of Carrie's two displacements using the bearing angles provided. Since bearings start from the north and move clockwise, S80°W corresponds to 100° measured counterclockwise from the east axis. The first displacement components can be found using sine and cosine functions:

• First westward component = 1.5 miles × cos(100°)
• First southward component = 1.5 miles × sin(100°)

After the change in bearing to S19°W (or 161° from the east axis), we calculate the second displacement components similarly:

• Second westward component = 3 miles × cos(161°)
• Second southward component = 3 miles × sin(161°)

By adding the respective components, we can form the total westward and southward displacement vectors. The magnitude of Carrie's resultant displacement from the lodge can then be found using the Pythagorean theorem applied to these total components:

1. Calculate the sum of the westward components (total westward displacement).
2. Calculate the sum of the southward components (total southward displacement).
3. The distance from the lodge = √(total westward displacement² + total southward displacement²).

This will give us the straight-line distance from the lodge. To find the direction, we can calculate the bearing from the lodge to Carrie's final position using arctangent on the ratio of the total southward displacement to the total westward displacement.