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To solve the problem, let's analyze the information given in the diagram and apply trigonometric concepts to find the angle that Devora turned.

### Given:
- The distance from the cave entrance to the empty chest is \( 48 \, \text{m} \).
- The distance from the cave entrance to the treasure is \( 95 \, \text{m} \).
- The angle between the line connecting the cave entrance to the chest and the line connecting the cave entrance to the treasure is \( 30^\circ \).

### To Find:
- The angle that Devora turned (which we’ll denote as \( \theta \)).

### Solution:
We need to find \( \theta \), the angle between the path from the chest to the treasure. We can use the **Law of Cosines** to find \( \theta \) in the triangle formed by the entrance, the chest, and the treasure.

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Where:
- \( a = 48 \, \text{m} \) (distance from entrance to chest)
- \( b = 95 \, \text{m} \) (distance from entrance to treasure)
- \( C = 30^\circ \) (angle between these two distances)
- \( c \) is the distance between the chest and the treasure, which we need to calculate first.

1. **Calculate \( c \) (distance between the chest and the treasure):**
   \[
   c^2 = 48^2 + 95^2 - 2 \times 48 \times 95 \times \cos(30^\circ)
   \]
   Let's calculate this.

2. **Solve for \( \theta \):**
   Once \( c \) is known, we can use the Law of Cosines again to find \( \theta \) in the triangle formed by the chest, entrance, and treasure:
   \[
   \cos(\theta) = \frac{a^2 + c^2 - b^2}{2ac}
   \]
   After calculating, we'll take the arccosine to find \( \theta \).

Let's calculate this step by step. I'll begin with the calculation of \( c \).The distance between the chest and the treasure, \( c \), is approximately \( 58.57 \, \text{m} \).

Now, let's use the Law of Cosines again to find the angle \( \theta \), the angle Devora turned. We will solve for \( \theta \) using the following formula:

\[
\cos(\theta) = \frac{a^2 + c^2 - b^2}{2ac}
\]

I'll calculate \( \theta \) next.The calculated angle \( \theta \) is approximately \( 125.81^\circ \). However, since the problem specifies that the angle Devora turned should be acute, we need to consider the acute angle, which would be:

\[
\text{Acute } \theta = 180^\circ - 125.81^\circ = 54.19^\circ
\]

Therefore, Devora turned approximately **54.19 degrees**.

Would you like any further details, or do you have any additional questions?

Here are 5 related questions you might find interesting:
1. How does the Law of Cosines help in solving triangles that aren't right-angled?
2. What are some common applications of the Law of Cosines in real-world problems?
3. How would the problem change if the angle between the chest and the treasure was obtuse?
4. What are the different ways to solve for the missing side of a triangle when given two sides and an included angle?
5. How does the angle of deviation relate to the problem of finding the turning angle in a pathfinding scenario?

**Tip:** When dealing with triangle problems, drawing a diagram with all known sides and angles clearly labeled can significantly simplify the process of identifying the correct formulas to use.

Devora explored a secret cave. 48 m from the entrance, she found an empty chest with a map drawn on it. The map showed treasure buried 95 m from the entrance to the cave. From the treasure, the map showed a 30° angle between the chest and the entrance. Devora faced the entrance, then turned a certain number of degrees to her left and walked until she got to the treasure. Supposing that the angle Devora turned is acute, how many degrees did Devora turn?

Solve a trigonometry problem involving the Law of Cosines to find the acute angle Devora turned in a cave exploration scenario. The problem provides key data points, including distances and an angle, making it suitable for high school students in Grades 9-12.

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