Certainly! Let's solve the problem step-by-step to find the direction of the man's resultant vector given that he walks 500 feet west and then 371 feet at an angle of [tex]\(52^\circ\)[/tex] south of west.
1. Set up the problem with components:
- When the man walks 500 feet west, he travels purely in the west direction.
- For the second part of his walk, he travels at an angle of [tex]\(52^\circ\)[/tex] south of west for 371 feet.
2. Convert the angle to the usual coordinate system:
- South of west means we can break this into west and south components.
- Let's decompose the 371 feet into westward and southward components using trigonometry.
3. Calculate the westward (x) and southward (y) components:
- For the westward component:
[tex]\[
\text{West component} = 371 \cdot \cos(52^\circ)
\][/tex]
- For the southward component:
[tex]\[
\text{South component} = 371 \cdot \sin(52^\circ)
\][/tex]
4. Sum the total westward and southward components:
- Total westward distance:
[tex]\[
R_x = 500 + 371 \cdot \cos(52^\circ)
\][/tex]
- Total southward distance:
[tex]\[
R_y = 371 \cdot \sin(52^\circ)
\][/tex]
5. Use these components to determine the angle [tex]\( \theta \)[/tex] of the resultant vector:
- The angle [tex]\( \theta \)[/tex] with respect to the west direction (clockwise from west):
[tex]\[
\theta = \arctan\left(\frac{R_y}{R_x}\right)
\][/tex]
- To find the direction bearing from the west:
[tex]\[
\theta = 180^\circ - \theta
\][/tex]
6. Calculate the direction:
The detailed steps give us the value:
[tex]\[
\theta = 158.13^\circ
\][/tex]
So, after rounding to the nearest hundredth, the direction of the man's resultant vector is:
[tex]\[
\boxed{158.13^\circ}
\][/tex]
