Pall walks 4 m to the west, then 8 m north, and lastly 6 m east. What is his resultant displacement?

We can solve this problem by determining Pall's final position relative to his starting point, which will give us his resultant displacement.

### Step 1: Break the movement into coordinates
Assume Pall starts at the origin \((0, 0)\).
- First, he walks 4 m west, which is in the negative x-direction. His new position is \((-4, 0)\).
- Next, he walks 8 m north, which is in the positive y-direction. His new position is \((-4, 8)\).
- Lastly, he walks 6 m east, which is in the positive x-direction. His new position is \((2, 8)\).

### Step 2: Use the Pythagorean theorem to find the resultant displacement
The displacement is the straight-line distance from the starting point \((0, 0)\) to the final position \((2, 8)\). We can calculate this using the formula for the distance between two points:

\[
\text{Displacement} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (2, 8)\):

\[
\text{Displacement} = \sqrt{(2 - 0)^2 + (8 - 0)^2}
\]
\[
\text{Displacement} = \sqrt{2^2 + 8^2}
\]
\[
\text{Displacement} = \sqrt{4 + 64} = \sqrt{68}
\]

\[
\text{Displacement} \approx 8.25 \, \text{m}
\]

### Step 3: Direction of the displacement
The angle of the displacement can be found using the inverse tangent function:

\[
\theta = \tan^{-1} \left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1} \left(\frac{8}{2}\right) = \tan^{-1}(4)
\]

\[
\theta \approx 75.96^\circ \, \text{north of east}.
\]

Thus, Pall's resultant displacement is approximately 8.25 meters at an angle of \(75.96^\circ\) north of east.

Do you need more details or have any questions?

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Here are 5 related questions:
1. What if Pall walked 10 m south instead of 8 m north?
2. How would the displacement change if Pall walked west first, then north and south?
3. Can you find Pall's average velocity if this trip took 10 seconds?
4. What would be Pall’s resultant displacement if he also walked 5 m south after his last movement?
5. How do we calculate displacement in three-dimensional movement?

**Tip**: When dealing with displacement, always consider the direction since displacement is a vector quantity!

Find Pall's resultant displacement after walking 4 m west, 8 m north, and 6 m east. Learn how to apply the Pythagorean theorem and trigonometric ratios to solve this vector problem. This solution is perfect for students in Grades 9-10 learning about displacement and direction.

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