Bear Grylls was dropped off in the wilderness and spots a camp 40degrees west of north. He proceeds to hike 15degrees east of north for 3.75 miles and determines the camp is now 80degrees west of north. How far is he from the camp at this point? Answer to the nearest hundredth of a mile.

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.

---

### 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 ,

Bear Grylls was dropped off in the wilderness and spots a camp 40 degrees west of north. He proceeds to hike 15 degrees east of north for 3.75 miles and determines the camp is now 80 degrees west of north. How far is he from the camp at this point? Answer to the nearest hundredth of a mile.

Solve a classic navigation problem involving Bear Grylls hiking in the wilderness. Learn how to apply vector analysis and the Law of Cosines to determine the distance to the camp after hiking 3.75 miles. This problem is ideal for students in Grades 10-12 and covers trigonometry concepts.

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