To determine how far Jane is from her starting point, we can treat this as a problem involving the Pythagorean theorem and trigonometry.

1. Understanding the Problem:

   • Jane starts walking North for 3 miles.
   • Then, she turns $45^\circ$ to the right, meaning she walks along a path that forms a $45^\circ$ angle with her initial northward path.
   • After turning, she walks 4 miles.

2. Visualizing the Movement:

   • Jane's walk can be broken down into two parts:
     1. First segment: She walks North for 3 miles.
     2. Second segment: After turning $45^\circ$ to the right, she walks 4 miles at a $45^\circ$ angle.

3. Using the Law of Cosines: To find the straight-line distance from her starting point, we can use the law of cosines. The law of cosines is a formula that relates the sides and angles of a triangle:

   $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ where:

   • $a = 3$ miles (the first segment),
   • $b = 4$ miles (the second segment),
   • $C = 45^\circ$ (the angle between the two segments),
   • $c$ is the distance from the starting point.

4. Substituting the values: $c^2 = 3^2 + 4^2 - 2(3)(4) \cdot \cos(45^\circ)$

   Since $\cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071$, we substitute that value in:

   $c^2 = 9 + 16 - 2(3)(4) \cdot 0.7071$ $c^2 = 25 - 24 \cdot 0.7071$ $c^2 = 25 - 16.9704$ $c^2 = 8.0296$ $c = \sqrt{8.0296} \approx 2.83$

Thus, Jane is approximately 2.83 miles from her starting point.

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Would you like more details on the solution process, or do you have any other questions?

Here are five related questions:

1. How can the law of sines be applied to a similar problem?
2. What happens if Jane turns a different angle, say $30^\circ$?
3. How can we calculate the displacement when the walking path forms a straight line?
4. What is the difference between using the law of cosines and the Pythagorean theorem?
5. How do you calculate the total distance traveled along a curved path?

Tip: When using the law of cosines, remember to check whether the angle between the two sides is given or needs to be calculated!