Question

In the packaging part of a warehouse, boxes of blue cups must be moved from the staging area located at an origination point along a conveyor belt described by a vector with components ⟨6, 30⟩. If boxes of red cups are moved 2.5 times the distance along the same belt, which describes the vector that shows the path of the boxes of red cups? Round the magnitude to the nearest whole number if necessary.

1) components: ⟨2.4, 12⟩, magnitude: about 31
2) components: ⟨2.4, 12⟩, magnitude: about 76
3) components: ⟨15, 75⟩, magnitude: about 31
4) components: ⟨15, 75⟩, magnitude: about 76

Answer 1

To find the vector for the path of the boxes of red cups, we multiply each component of the blue cups' vector by 2.5, yielding the new vector ➤15, 75➤. Calculating the magnitude with the Pythagorean theorem gives us a value of approximately 76.

Step-by-step explanation:

The question involves determining the vector that represents the path of boxes of red cups along a conveyor belt, given that they travel 2.5 times the distance of boxes of blue cups, which are moved along the vector with components ➤6, 30➤. To find the new vector for the red cups, we simply multiply each component of the blue cups' vector by 2.5. The components are calculated as 2.5 × 6 = 15 and 2.5 × 30 = 75, giving us the new vector components ➤15, 75➤ for the red cups. To find the magnitude of the vector, we use the Pythagorean theorem which gives √(15² + 75²) = √(225 + 5625) = √5850, which rounds to about 76 when rounded to the nearest whole number.

Answer 2

Components: ⟨15, 75⟩, Magnitude: about 76 signifies the total distance traveled by the red cups, confirming the significant increase due to the 2.5 times factor.

Explanation:

The vector for the red cups' path is described by components ⟨15, 75⟩, indicating that the boxes of red cups move 2.5 times the distance of the blue cups' path.

The magnitude, about 76, aligns with this, representing the length of the vector, rounded to the nearest whole number. This indicates a significant increase from the original distance traveled by the blue cups along the conveyor belt, reflecting the 2.5 times factor.

The vector components ⟨15, 75⟩ signify that for every unit moved in the horizontal direction (x-axis), the red cups move 5 times that distance along the vertical direction (y-axis), portraying the 2.5 times increase compared to the blue cups' movement.

The magnitude of approximately 76 signifies the total distance traveled by the red cups, confirming the significant increase due to the 2.5 times factor.