Math Problem Statement
If Jane walks North for 3 miles, turns 45 degrees to the right, and then walks another 4 miles, how many miles will Jane be from her starting point? Give your answer as a decimal rounded to the nearest hundredth.
Solution
This problem can be solved using trigonometry and the Pythagorean theorem, since Jane's movements form a triangle.
- First leg of the journey: Jane walks 3 miles north.
- Second leg of the journey: After turning 45 degrees to the right, Jane walks 4 miles.
The two legs of her journey form two sides of a triangle, with the angle between them being 45°. We can now use the law of cosines to find the distance from Jane's starting point.
The law of cosines formula is: Where:
- miles (first leg),
- miles (second leg),
- (the angle between the two legs).
Now, plug in the values:
So, Jane is approximately 2.83 miles from her starting point.
Would you like more details or have any questions?
Here are 5 questions you can explore:
- How would the result change if the second leg was 5 miles instead of 4 miles?
- What if Jane turned 60 degrees instead of 45 degrees after walking north?
- How would we solve the problem if Jane walked east instead of north at the start?
- Can you derive the law of cosines for any general triangle?
- How can this problem be solved using a vector approach?
Tip: For non-right triangles, always check if the law of cosines or sines is more suitable.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Geometry
Law of Cosines
Pythagorean Theorem
Formulas
c^2 = a^2 + b^2 - 2ab * cos(θ)
Theorems
Law of Cosines
Suitable Grade Level
Grades 10-12
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