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To determine the direction of the taxi's resultant vector, let's break the problem down into a series of steps. The taxi drives 6 miles east, then turns and drives at an angle of [tex]\(45^{\circ}\)[/tex] south of east for another 6 miles.
### Step-by-Step Solution:
1. Convert the angle to radians.
The angle given is [tex]\(45^{\circ}\)[/tex]. We need to convert this angle to radians because trigonometric functions in standard mathematical libraries typically use radians.
[tex]\[ \text{Angle in radians} = \frac{45 \times \pi}{180} = \frac{\pi}{4} \][/tex]
This results in approximately [tex]\(0.7854\)[/tex] radians.
2. Decompose the second leg of the trip into east and south components.
The second leg of the trip (6 miles at [tex]\(45^{\circ}\)[/tex] south of east) can be split into east and south components using sine and cosine functions.
[tex]\[ \text{East Component} = 6 \times \cos\left(\frac{\pi}{4}\right) \approx 6 \times 0.7071 \approx 4.2426\, \text{miles} \][/tex]
[tex]\[ \text{South Component} = 6 \times \sin\left(\frac{\pi}{4}\right) \approx 6 \times 0.7071 \approx 4.2426\, \text{miles} \][/tex]
3. Calculate the total distance traveled east and south.
The taxi initially drives 6 miles east and then an additional east component of the second leg.
[tex]\[ \text{Total East Distance} = 6 + 4.2426 \approx 10.2426\, \text{miles} \][/tex]
The total south distance is just the south component from the second leg.
[tex]\[ \text{Total South Distance} = 4.2426\, \text{miles} \][/tex]
4. Determine the angle of the resultant vector with respect to the east direction.
To find the resultant direction, we can use the inverse tangent (arctan) function:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Total South Distance}}{\text{Total East Distance}}\right) \][/tex]
Substituting the values obtained:
[tex]\[ \theta = \tan^{-1}\left(\frac{4.2426}{10.2426}\right) \approx \tan^{-1}(0.4142) \approx 22.5^{\circ} \][/tex]
Therefore, the angle of the taxi's resultant vector with respect to the east is approximately [tex]\(22.5^{\circ}\)[/tex].
Rounding to the nearest hundredth:
[tex]\[ \boxed{22.50^{\circ}} \][/tex]
Thus, the direction of the taxi's resultant vector is [tex]\(22.50^{\circ}\)[/tex] south of east.
### Step-by-Step Solution:
1. Convert the angle to radians.
The angle given is [tex]\(45^{\circ}\)[/tex]. We need to convert this angle to radians because trigonometric functions in standard mathematical libraries typically use radians.
[tex]\[ \text{Angle in radians} = \frac{45 \times \pi}{180} = \frac{\pi}{4} \][/tex]
This results in approximately [tex]\(0.7854\)[/tex] radians.
2. Decompose the second leg of the trip into east and south components.
The second leg of the trip (6 miles at [tex]\(45^{\circ}\)[/tex] south of east) can be split into east and south components using sine and cosine functions.
[tex]\[ \text{East Component} = 6 \times \cos\left(\frac{\pi}{4}\right) \approx 6 \times 0.7071 \approx 4.2426\, \text{miles} \][/tex]
[tex]\[ \text{South Component} = 6 \times \sin\left(\frac{\pi}{4}\right) \approx 6 \times 0.7071 \approx 4.2426\, \text{miles} \][/tex]
3. Calculate the total distance traveled east and south.
The taxi initially drives 6 miles east and then an additional east component of the second leg.
[tex]\[ \text{Total East Distance} = 6 + 4.2426 \approx 10.2426\, \text{miles} \][/tex]
The total south distance is just the south component from the second leg.
[tex]\[ \text{Total South Distance} = 4.2426\, \text{miles} \][/tex]
4. Determine the angle of the resultant vector with respect to the east direction.
To find the resultant direction, we can use the inverse tangent (arctan) function:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Total South Distance}}{\text{Total East Distance}}\right) \][/tex]
Substituting the values obtained:
[tex]\[ \theta = \tan^{-1}\left(\frac{4.2426}{10.2426}\right) \approx \tan^{-1}(0.4142) \approx 22.5^{\circ} \][/tex]
Therefore, the angle of the taxi's resultant vector with respect to the east is approximately [tex]\(22.5^{\circ}\)[/tex].
Rounding to the nearest hundredth:
[tex]\[ \boxed{22.50^{\circ}} \][/tex]
Thus, the direction of the taxi's resultant vector is [tex]\(22.50^{\circ}\)[/tex] south of east.
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