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Sagot :
Sure, let's break down the problem step-by-step.
### Part 1: Finding the Total Distance from Home
The woman starts her journey by walking 3 miles to the west, and then she walks 7 miles southwest. We need to determine her total distance from home.
1. Initial Movement:
- She walks 3 miles to the west.
2. Southwest Movement:
- Walking southwest means moving at a 45-degree angle from the due west direction.
- We can break this movement into two components: one westward and one southward.
- Since the angle forming the southwest direction is 45 degrees, the westward and southward components of the 7-mile walk can be calculated using trigonometric functions:
- The westward component is [tex]\(7 \times \cos(45^\circ)\)[/tex].
- The southward component is [tex]\(7 \times \sin(45^\circ)\)[/tex].
- Both [tex]\(\cos(45^\circ)\)[/tex] and [tex]\(\sin(45^\circ)\)[/tex] equal [tex]\(\frac{\sqrt{2}}{2}\)[/tex], which is approximately 0.707.
3. Total Displacement:
- Calculate the total westward movement:
[tex]\[ \text{Total westward movement} = 3 + 7 \times 0.707 \approx 3 + 4.949 \approx 7.949 \text{ miles} \][/tex]
- Calculate the total southward movement:
[tex]\[ \text{Total southward movement} = 7 \times 0.707 \approx 4.949 \text{ miles} \][/tex]
4. Total Distance:
- To find the total distance from her starting point, we use the Pythagorean theorem. The distance [tex]\(D\)[/tex] from home is given by:
[tex]\[ D = \sqrt{(\text{westward distance})^2 + (\text{southward distance})^2} = \sqrt{(7.949)^2 + (4.949)^2} \approx 9.365 \text{ miles} \][/tex]
Therefore, the woman is approximately 9.365 miles from her home.
### Part 2: Finding the Direction to Head Directly Home
To determine the direction she must walk to head directly home, we need to find the angle relative to north of east.
1. Calculate the angle to home:
- The angle can be calculated using the tangent function:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{southward distance}}{\text{westward distance}}\right) = \tan^{-1}\left(\frac{4.949}{7.949}\right) \approx 32.092^\circ \text{ south of west} \][/tex]
2. Convert to north of east:
- Since the angle calculated is south of west, to convert it to north of east, we subtract this angle from 180 degrees:
[tex]\[ \text{Angle north of east} = 180^\circ - 32.092^\circ = 147.908^\circ \][/tex]
Therefore, the woman must head approximately 148.092 degrees north of east to walk directly home.
### Part 1: Finding the Total Distance from Home
The woman starts her journey by walking 3 miles to the west, and then she walks 7 miles southwest. We need to determine her total distance from home.
1. Initial Movement:
- She walks 3 miles to the west.
2. Southwest Movement:
- Walking southwest means moving at a 45-degree angle from the due west direction.
- We can break this movement into two components: one westward and one southward.
- Since the angle forming the southwest direction is 45 degrees, the westward and southward components of the 7-mile walk can be calculated using trigonometric functions:
- The westward component is [tex]\(7 \times \cos(45^\circ)\)[/tex].
- The southward component is [tex]\(7 \times \sin(45^\circ)\)[/tex].
- Both [tex]\(\cos(45^\circ)\)[/tex] and [tex]\(\sin(45^\circ)\)[/tex] equal [tex]\(\frac{\sqrt{2}}{2}\)[/tex], which is approximately 0.707.
3. Total Displacement:
- Calculate the total westward movement:
[tex]\[ \text{Total westward movement} = 3 + 7 \times 0.707 \approx 3 + 4.949 \approx 7.949 \text{ miles} \][/tex]
- Calculate the total southward movement:
[tex]\[ \text{Total southward movement} = 7 \times 0.707 \approx 4.949 \text{ miles} \][/tex]
4. Total Distance:
- To find the total distance from her starting point, we use the Pythagorean theorem. The distance [tex]\(D\)[/tex] from home is given by:
[tex]\[ D = \sqrt{(\text{westward distance})^2 + (\text{southward distance})^2} = \sqrt{(7.949)^2 + (4.949)^2} \approx 9.365 \text{ miles} \][/tex]
Therefore, the woman is approximately 9.365 miles from her home.
### Part 2: Finding the Direction to Head Directly Home
To determine the direction she must walk to head directly home, we need to find the angle relative to north of east.
1. Calculate the angle to home:
- The angle can be calculated using the tangent function:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{southward distance}}{\text{westward distance}}\right) = \tan^{-1}\left(\frac{4.949}{7.949}\right) \approx 32.092^\circ \text{ south of west} \][/tex]
2. Convert to north of east:
- Since the angle calculated is south of west, to convert it to north of east, we subtract this angle from 180 degrees:
[tex]\[ \text{Angle north of east} = 180^\circ - 32.092^\circ = 147.908^\circ \][/tex]
Therefore, the woman must head approximately 148.092 degrees north of east to walk directly home.
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