Question

An adventurous dog strays from home, runs three blocks east, two blocks north, one block east, one block north, and two blocks west. Assuming that each block is 104 m. How far from home is the dog, in units of m? Also what direction is the dog? Express the direction as an < measured in degress north of east.

Answer 1

The adventurous dog is approximately 375 meters from home at an angle of about 56° north of east after accounting for all the blocks traveled in the respective directions.

Step-by-step explanation:

To determine how far the dog is from home and in what direction, we add the vectors representing the dog's movements. The dog runs three blocks east, two blocks north, one block east, one block north, and finally two blocks west. Each block is 104 meters. We can add these displacements using vectors:

1. East: 3 + 1 - 2 blocks = 2 blocks
2. North: 2 + 1 block = 3 blocks

The total displacement in meters for the east and north directions are:

1. East: 2 blocks × 104 m/block = 208 m
2. North: 3 blocks × 104 m/block = 312 m

We find the resultant vector's magnitude using the Pythagorean theorem:

√(208^2 + 312^2) = √(43264 + 97344) = √140608 ≈ 374.98 m

The direction is given by the angle θ north of east, which we find using the tangent function:

tan(θ) = north/east = 312 m / 208 m = 1.5

θ = tan^{-1}(1.5) ≈ 56.31°

Thus, the dog is approximately 375 m from home at an angle of about 56° north of east. If the blocks were 100 m instead, using the same calculations, the distance would remain close to the same value, and the direction would not change.