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To understand how the girl’s route translates into a single movement, we use the concept of vector addition. Imagine each segment of her journey as a vector, which indicates both direction and magnitude. This is a common way to describe displacement in physics.

The girl travels in three directions: west, north, and east.

• The westward movement can be seen as a vector pointing to the left.
• Northward movement is a vertical vector pointing upwards.
• Eastward movement is a vector pointing to the right.

These vectors can be added together to find a resultant vector, which shows the direct path from her starting point to her ending point. By breaking each vector down into components (such as just horizontal or vertical movement), we can simplify the math and find the end result of her combined movements. In this exercise, the horizontal (east-west) vectors are combined first, followed by the inclusion of the vertical (north) vector to determine the overall displacement.

Once the vector components are sorted, the Pythagorean theorem comes into play. This mathematical principle is crucial for finding the magnitude of the resultant vector (displacement) in problems involving right-angled triangles.

In simple terms, if you have two sides of a right-angle triangle, you can find the third side (the hypotenuse) using the formula: \[ c = \sqrt{a^2 + b^2} \] Where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. For the newspaper delivery girl, her path forms such a triangle:

• A (east-west) side of \(3.00\) blocks.
• A vertical (north) side of \(4.00\) blocks.

By applying the theorem, \(c\), her displacement, is \(5.00\) blocks. This shows the beauty of the Pythagorean theorem in calculating direct paths from journey splits.

Understanding the concept of distance calculation is fundamental when determining the total path someone has traveled. Unlike displacement, which is concerned with the most direct route from start to finish, total distance focuses on every segment covered. In this case, the girl’s journey isn't a straight line.

Her total distance is simply the sum of all individual segments of her trip. Here's how it's calculated:

• She travels \(3.00\) blocks west.
• Then \(4.00\) blocks north.
• Finally, \(6.00\) blocks east.

Adding these values gives \(13.00\) blocks as her total traveled distance.

While displacement asks "how far out of place" the girl is, the total distance asks "how far has she walked in all?" It’s a measure that doesn’t care about direction, only total ground covered.