In the diagram ​ below, what is the relationship between the number of hexagons and the perimeter of the figure they ​ form? Represent this relationship using a ​ table, words, an ​ equation, and a graph. Let x equals the number of hexagons. Let y equals the perimeter of the figure.

Question

2
2
2
2
1 hexagon
2 hexagons
3 hexagons
2
2
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Hexagons​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
  
enter your response here
  
enter your response here
2
  
enter your response here
  
enter your response here
3
  
enter your response here
  
enter your response here
​(Simplify your​ answers.)

Answer 1

To describe the relationship between the number of hexagons and the perimeter, we can consider the arrangement of hexagons. When hexagons are placed together in a flat arrangement, they share sides, which reduces the total perimeter.

1. Understanding the Perimeter of a Hexagon:

   • A single hexagon has a perimeter of \(6s\), where \(s\) is the length of one side of the hexagon. Assuming each hexagon is identical and side length \(s = 2\), the perimeter of a single hexagon would be \(6 \cdot 2 = 12\).

2. Hexagons arrangement:

   • When hexagons are arranged:
     • 1 hexagon: The perimeter is simply the perimeter of one hexagon.
     • 2 hexagons: They will share one side, so the perimeter would be reduced by 2 (one side each).
     • 3 hexagons: They may share sides based on how they're arranged. In a straight line, they will share two sides.

Table Representation

Calculating the perimeter based on the arrangement:

• For \(x = 1\) hexagon: \(y = 12\)
• For \(x = 2\) hexagons: \(y = 12 - 2 = 10\)
• For \(x = 3\) hexagons: \(y = 12 - 2 - 2 = 8\) (based on a line arrangement)

| Number of Hexagons, x | Perimeter, y | Ordered Pair (x, y) | |------------------------|---------------|----------------------| | 1 | 12 | (1, 12) | | 2 | 10 | (2, 10) | | 3 | 8 | (3, 8) |

Words Representation

The perimeter of the figure formed by hexagons decreases as more hexagons are added. Specifically, when the hexagons are placed adjacent to one another, each new hexagon added reduces the perimeter by a certain amount due to shared sides.

Equation Representation

From our findings, we can derive a linear equation representing the perimeter based on the number of hexagons:

• If \(y = 12 - 2(x - 1)\), where \(2(x - 1)\) accounts for the sides shared as more hexagons are added.

So, if simplified:

• \(y = 14 - 2x\)

Graph Representation

To graph the relationship:

• Plot the points (1, 12), (2, 10), (3, 8) on a coordinate plane, where the x-axis represents the number of hexagons, and the y-axis represents the perimeter. Draw a line connecting the points, which will show a linear decrease in the perimeter as the number of hexagons increases.

This describes the relationship comprehensively!

Answer 2

In the diagram​ below, what is the relationship between the number of hexagons and the perimeter of the figure they​ form? Represent this relationship using a​ table, words, an​ equation, and a graph. Let x equals the number of hexagons. Let y equals the perimeter of the figure.
2
2
2
2
1 hexagon
2 hexagons
3 hexagons
2
2
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Hexagons​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
enter your response here
  
enter your response here
2
  
enter your response here
  
enter your response here
3
  
enter your response here
  
enter your response here
​(Simplify your​ answers.)

Answer 3

To analyze the relationship between the number of hexagons and the perimeter of the figure they form, we can first define a few key characteristics of hexagons and how they interact when arranged together.

Understanding the Hexagon Perimeter:

1. Perimeter of a single hexagon: Standard hexagon has 6 sides. If each side has a length of \(s\) (let's say \(s = 2\)), the perimeter of one hexagon would be: \[ P = 6 \times 2 = 12 \]

2. Adjacent hexagons: Each time a new hexagon is added, it will share sides with the existing hexagons, which decreases the total perimeter.

Calculating the Perimeter:

• 1 hexagon: The perimeter is \(12\).
• 2 hexagons: When arranged side by side, they will share one side, so the perimeter will be: \[ 12 - 2 = 10 \]
• 3 hexagons: Arranged in a straight line, they will share two sides, leading to: \[ 12 - 2 \times 2 = 8 \]

Relationship Table

Now let's fill out the table with the values computed:

| Number of Hexagons​, x | Perimeter, y | Ordered Pair (x, y) | |------------------------|---------------|----------------------| | 1 | 12 | (1, 12) | | 2 | 10 | (2, 10) | | 3 | 8 | (3, 8) |

Words Representation

The perimeter of the figure formed by hexagons decreases as more hexagons are added. When additional hexagons are connected to the existing ones, they share sides, leading to a reduction in the overall perimeter.

Equation Representation

From our findings, we can derive the following linear equation:

• The equation based on the relationship can be:

\[ y = 14 - 2x \]

Where:

• \(y\) is the perimeter,
• \(x\) is the number of hexagons.

Graph Representation

To graph the relationship:

• Plot the points (1, 12), (2, 10), and (3, 8) on a coordinate plane.
• The x-axis represents the number of hexagons, while the y-axis represents the perimeter.
• Draw a line connecting these points, showing that as the number of hexagons increases, the perimeter decreases.

This summarizes the relationship between the number of hexagons and the perimeter of the figure they form! If you need assistance with the graph or any further details, feel free to ask!