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We are given a diagram showing a hexagonal tile, and we are asked to find \( x \) when the area of the hexagon is \( 18\sqrt{3} \, \text{in}^2 \).

To solve this, let's break it down step by step.

### Step 1: Formula for the Area of a Regular Hexagon
The area \( A \) of a regular hexagon with side length \( s \) is given by:

\[
A = \frac{3\sqrt{3}}{2} s^2
\]

We know from the problem that the area is \( 18\sqrt{3} \). Thus, we can set up the equation:

\[
18\sqrt{3} = \frac{3\sqrt{3}}{2} s^2
\]

### Step 2: Solving for the Side Length \( s \)
We can simplify the equation by dividing both sides by \( \sqrt{3} \):

\[
18 = \frac{3}{2} s^2
\]

Multiply both sides by 2 to eliminate the fraction:

\[
36 = 3 s^2
\]

Now, divide by 3:

\[
s^2 = 12
\]

Take the square root of both sides:

\[
s = \sqrt{12} = 2\sqrt{3}
\]

### Step 3: Relating \( s \) to \( x \)
From the diagram, it looks like the side length \( s \) of the hexagon is related to the expression involving \( x \). In particular, the side length \( s \) appears to be \( \frac{x}{6} \sqrt{3} \) based on the labels given in the diagram.

Thus, we equate:

\[
s = \frac{x \sqrt{3}}{6}
\]

Substitute \( s = 2\sqrt{3} \) from the previous result:

\[
2\sqrt{3} = \frac{x \sqrt{3}}{6}
\]

### Step 4: Solving for \( x \)
We can divide both sides by \( \sqrt{3} \) to get rid of the square roots:

\[
2 = \frac{x}{6}
\]

Multiply both sides by 6:

\[
x = 12
\]

Thus, \( x = 12 \).

Would you like more details or have any questions?

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Here are 5 related questions that expand upon this problem:
1. How do we derive the formula for the area of a regular hexagon?
2. If the area of the hexagon were different, how would the value of \( x \) change?
3. How do the geometric properties of hexagons relate to other polygons?
4. What other methods could we use to calculate the side length of a hexagon?
5. How can this type of problem be extended to three-dimensional shapes like a hexagonal prism?

### Tip:
When solving geometry problems involving regular polygons, remember to relate the side lengths and areas through known formulas to simplify your calculations.

The diagram shows a hexagon-shaped tile used for flooring. Each hexagon tile has an area of 18√3 in². Find x.

Learn how to find the value of x in a hexagonal tile problem, where the tile's area is 18√3 in². This problem involves geometry, algebra, and step-by-step reasoning using the formula for the area of a regular hexagon.

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