We can solve this problem using vector analysis and trigonometry. This is a classic navigation problem, where we need to find the distance between two points based on directions and displacements.

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Step 1: Define the directions and positions

• Initial Camp Bearing: 40° west of north.
• Bear's Hike Direction: 15° east of north.
• New Camp Bearing: 80° west of north.
• Distance Hiked: 3.75 miles.

We need to find the distance from Bear’s final position to the camp. This is a problem involving the law of cosines, as the positions of the camp and Bear form a triangle with known angles and side lengths.

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Step 2: Draw or visualize the triangle

1. Let Bear's starting point be $A$.
2. Let the camp be $C$.
3. Let Bear's current position after hiking be $B$.

We need to find the distance $d = BC$, which is Bear's distance to the camp from his new location.

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Step 3: Use relative angles

Since directions are given relative to north, we need to compute the angle between the two bearings (the camp's directions before and after hiking).

• Initial camp bearing: 40° west of north.
• New camp bearing: 80° west of north.

Thus, the angle between these two directions is:

$80^\circ - 40^\circ = 40^\circ$

This 40° is the included angle in our triangle.

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Step 4: Apply the Law of Cosines

The Law of Cosines states:

$c^2 = a^2 + b^2 - 2ab \cos(\theta)$

Where:

• $a = 3.75$ miles (the distance Bear hiked)
• $b = d = BC$ (the distance to the camp, which we need to find)
• $\theta = 40^\circ$ (the angle between the two directions)

So:

[ d^2 = 3.75^2 + b^2 ,