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### Problem Description:
- There are three points: Ana (A), Carlos (C), and Betty (B).
- Ana and Carlos are separated by a distance of \( \sqrt{7} \) meters.
- Carlos and Betty are separated by \( 2\sqrt{2} \) meters.
- Betty and Ana are separated by \( 3 \) meters.
- The angle of interest is \( \angle ACB \), and the task is to find the measure of this angle.

### Solution Strategy:
We can solve this using the **Cosine Rule**. The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of its angles:

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

Where:
- \( a \), \( b \), and \( c \) are the sides of the triangle,
- \( \theta \) is the angle opposite side \( c \).

Here:
- \( a = \sqrt{7} \) meters (distance between Ana and Carlos),
- \( b = 2\sqrt{2} \) meters (distance between Carlos and Betty),
- \( c = 3 \) meters (distance between Ana and Betty).

### Calculation:
We want to find \( \cos(\angle ACB) \):

\[
\cos(\angle ACB) = \frac{(\sqrt{7})^2 + (2\sqrt{2})^2 - 3^2}{2 \times \sqrt{7} \times 2\sqrt{2}}
\]

Substitute the values:

\[
\cos(\angle ACB) = \frac{7 + 8 - 9}{2 \times \sqrt{7} \times 2\sqrt{2}}
\]
\[
\cos(\angle ACB) = \frac{6}{4\sqrt{14}}
\]
\[
\cos(\angle ACB) = \frac{3}{2\sqrt{14}}
\]

Now let's simplify this to find the angle:

We convert \( \cos(\angle ACB) = \frac{3}{2\sqrt{14}} \) to an approximate value:

\[
\cos(\angle ACB) \approx 0.1601
\]

Finally, calculate the angle \( \angle ACB \):

\[
\angle ACB \approx \cos^{-1}(0.1601) \approx 80.8^\circ
\]

### Final Answer:
The closest option to this value is **d) 81°**.

Would you like further details or have any questions?

### Relative Questions:
1. How would the problem change if the side lengths were given as whole numbers?
2. What is the significance of the cosine rule in determining triangle angles?
3. If \( \angle ACB \) were \( 60^\circ \), how would you verify the side lengths?
4. What other trigonometric rules could be used if this was a right triangle?
5. How would you approach the problem if only the angles were given and not the sides?

### Tip:
When dealing with trigonometric problems, ensure your calculator is in the correct mode (degrees or radians) to avoid errors in angle calculations.

Learn how to calculate the angle ACB in a triangle using the Cosine Rule. Solve a geometric problem involving distances between points Ana, Carlos, and Betty. Detailed step-by-step solution provided with calculations and final angle determination.

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