To find the direction of the dog's resultant vector after running 250 feet south and then 252 feet at an angle of [tex]\( 25^{\circ} \)[/tex] west of south, we can follow these steps:
1. Determine the position of the second leg of the run:
- The dog first runs 250 feet directly south. This initial movement is a straightforward downward motion in the south direction.
- Next, the dog turns and runs 252 feet at an angle of [tex]\( 25^{\circ} \)[/tex] west of south.
2. Break down the second leg into its components:
- To analyze this motion, let's convert the angle to Cartesian coordinates. We will calculate the westerly (x) and southerly (y) components of this second part of the run.
- Westerly component ([tex]\( \Delta x \)[/tex]):
[tex]\[
\Delta x = 252 \cdot \sin(25^{\circ})
\][/tex]
- Southerly component ([tex]\( \Delta y \)[/tex]):
[tex]\[
\Delta y = 252 \cdot \cos(25^{\circ})
\][/tex]
3. Calculate the net displacements in both directions:
- The total westerly displacement ([tex]\( x \)[/tex]-direction) will simply be:
[tex]\[
\text{Total } x = -\Delta x
\][/tex]
Here, it's negative because moving west signifies a negative x-direction.
- The total southerly displacement ([tex]\( y \)[/tex]-direction) will be the sum of the initial 250 feet plus the southerly component of the second leg:
[tex]\[
\text{Total } y = -250 - \Delta y
\][/tex]
This remains negative due to the direction being south.
4. Determine the magnitude of the resultant vector:
- The magnitude of the resultant vector is found using the Pythagorean theorem:
[tex]\[
|\overrightarrow{R}| = \sqrt{(\text{Total } x)^2 + (\text{Total } y)^2}
\][/tex]
- Given [tex]\(|\overrightarrow{R}|\)[/tex] is 490.10 feet from our solution, we move to the next step.
5. Calculate the direction of the resultant vector:
- The direction angle [tex]\( \theta \)[/tex], measured relative to the south, can be found using the arctangent function:
[tex]\[
\theta = \tan^{-1}\left( \frac{\text{Total } x}{\text{Total } y} \right)
\][/tex]
- However, since it involves the correct mathematical correction:
[tex]\[
\theta = 180^\circ + \text{atan2}(\text{Total } x, \text{Total } y)
\][/tex]
From the given answer, the resultant angle relative to the south was calculated and concludes:
[tex]\[
\theta \approx 12.55^\circ
\][/tex]
Therefore, the direction of the dog's resultant vector is approximately [tex]\( 12.55^\circ \)[/tex] west of south.
