Math Problem Statement
A person leaves home and walks 2 miles west, then 5 miles southwest.
How far from home is she?
miles
In what direction must she walk to head directly home?
degrees North of East
Solution
To solve this problem, we can break the person's movement into two segments: the first segment (2 miles west) and the second segment (5 miles southwest). We can then use the Pythagorean theorem to find the straight-line distance from her final position to the starting point (home), and trigonometry to determine the direction she must head to return home.
Step 1: Break down the movements
- The first movement is 2 miles west. This is simply a horizontal displacement of 2 miles to the left (west).
- The second movement is 5 miles southwest. Southwest is at a 45-degree angle south of west, which means the movement forms a 45-degree angle with the horizontal axis. Using trigonometry, we can break this movement into two components:
- West component:
- South component:
Step 2: Calculate total displacement
- West component of the second movement:
- South component of the second movement:
Step 3: Total displacement in the west and south directions
- Total westward displacement: miles to the west
- Total southward displacement: 3.535 miles to the south
Step 4: Find the straight-line distance from home
To find the straight-line distance from her home (her total displacement), we use the Pythagorean theorem:
Step 5: Find the direction to head home
To find the direction, we use the inverse tangent function () to calculate the angle north of east: This means she must walk 32.5° east of north to head directly home.
Final Answer:
- Distance from home: miles
- Direction: North of East
Would you like a more detailed explanation of any step?
Related questions:
- How would the displacement change if she had walked only 3 miles southwest instead of 5?
- What is the formula for calculating the straight-line distance using the Pythagorean theorem?
- How can we adjust this solution if the person had walked 5 miles southeast instead of southwest?
- How can vector addition help in understanding the displacement?
- What would the result be if she had walked 2 miles north first instead of west?
Tip: Always break down the motion into components based on the direction (like north-south or east-west) before applying Pythagoras or trigonometric functions.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Pythagorean Theorem
Vector Addition
Formulas
Pythagorean Theorem: d = √(x² + y²)
Cosine and Sine for vector components: x = r * cos(θ), y = r * sin(θ)
Inverse Tangent: θ = tan⁻¹(y / x)
Theorems
Pythagorean Theorem
Trigonometric Functions
Vector Addition
Suitable Grade Level
Grades 9-12
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