A delivery truck travels 15 blocks north, 21 blocks east and 18 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.
magnitude and direction
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To find the final displacement from the origin, we need to calculate the magnitude (distance) and direction of the truck's final position.
1. Start by drawing a diagram to visualize the truck's movements. Let's assume the origin is at the center of the diagram.
- Start from the origin (center) and draw 15 blocks north (upwards).
- From the end of the previous line, draw 21 blocks east (to the right).
- From the end of the previous line, draw 18 blocks south (downwards).
2. Now, we need to calculate the magnitude of the displacement. To do that, we need to find the horizontal and vertical components of the displacement.
- Vertical component: Since the truck traveled 15 blocks north and then 18 blocks south, the net vertical displacement is 15 - 18 = -3 blocks south.
- Horizontal component: The truck traveled 21 blocks east, so the horizontal displacement is 21 blocks east.
3. Calculate the magnitude (distance) of the displacement using the Pythagorean theorem.
- Magnitude (distance) = √(horizontal displacement^2 + vertical displacement^2)
- = √(21^2 + (-3)^2)
- = √(441 + 9)
- = √450
- ≈ 21.213 blocks
4. Determine the direction of the displacement.
- To find the direction, we can use trigonometry. We can calculate the arctangent of the vertical displacement divided by the horizontal displacement.
- Direction = arctan(vertical displacement / horizontal displacement)
- = arctan(-3 / 21)
- ≈ -8.13 degrees (measured counterclockwise from the positive x-axis)
Therefore, the truck's final displacement from the origin is approximately 21.213 blocks in magnitude, in a direction of about -8.13 degrees counterclockwise from the positive x-axis.