You are on a treasure hunt and your map says "Walk due west for 52 paces, then walk 29.6° north of west for 40 paces, and finally walk due north for 25 paces." What is the magnitude of the component of your displacement in the following directions?

(a) due north
paces

(b) due west
paces

Label starting point (0,0)

after walking due west, you are at (-52,0)

From there, your next displacement is (40 cos 150.4° , 40 sin 150.4°) = (-34.78,7.22), and you are now at (-86.78,7.22)

From there, you now displace (0,25) and end up at (-86.78,32.22)

Another Method:

D = -52 + 40[150.4] + 25i
D = -52 + -34.8 + 19.8i + 25i
D = -86.8 + 44.8i, Q2.

D = (X,Y) = (-86.8, 44.8).

To find the magnitude of the component of your displacement, we can break down the given directions into their north and west components.

(a) Due North:
- From the given directions, we walked due north for 25 paces.
- The component of displacement in the north direction is simply 25 paces.
- Therefore, the magnitude of the component of your displacement in the due north direction is 25 paces.

(b) Due West:
- From the given directions, we walked due west for 52 paces.
- There is no component in the west direction from the second instruction since it is stated as a certain angle north of west.
- Therefore, the magnitude of the component of your displacement in the due west direction is 52 paces.

To find the magnitude of the component of displacement in a specific direction, we need to break down each step of the treasure hunt and calculate the displacement separately.

(a) To find the magnitude of the component due north, we need to calculate the displacement from the steps of walking due north. Let's break it down:

- The first step is to walk due west for 52 paces. Since we are walking due west, there is no displacement in the north direction.
- The second step is to walk 29.6° north of west for 40 paces. To find the northward component, we need to calculate the northward displacement introduced by this step.
- The angle is given as 29.6° north of west. Since west is the negative x-direction, we can calculate the northward component using trigonometry.
- The northward component = 40 paces * sin(29.6°) ≈ 19.32 paces.

- The final step is to walk due north for 25 paces. This step directly adds to the northward displacement.

To find the total magnitude of the northward displacement, we add up each component: 0 paces (due west) + 19.32 paces (29.6° north of west) + 25 paces (due north) = 44.32 paces.

Therefore, the magnitude of the component of your displacement due north is approximately 44.32 paces.

(b) To find the magnitude of the component due west, we need to calculate the displacement from the steps of walking due west. Let's break it down:

- The first step is to walk due west for 52 paces. This step contributes directly to the westward displacement.

To find the total magnitude of the westward displacement, we only consider the displacement due west: 52 paces.

Therefore, the magnitude of the component of your displacement due west is 52 paces.