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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 xequalsthe number of hexagons. Let yequalsthe 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
  
12
  
left parenthesis 1 comma 12 right parenthesis
2
  
22
  
left parenthesis 2 comma 22 right parenthesis
3
  
32
  
left parenthesis 3 comma 32 right parenthesis
​(Simplify your​ answers.)

1 answer

To analyze the relationship between the number of hexagons and the perimeter of the figure they form, let’s first complete the table by calculating the perimeter (y) for each number of hexagons (x).

Assumptions:

  • Each hexagon contributes a perimeter of 12 (if it's a regular hexagon).
  • When additional hexagons are added, the effective perimeter increase might not be linear (overlapping sides).

Commonly, for the arrangement of hexagons, the pattern for perimeter can be represented as each additional hexagon overlaps with existing hexagons, hence the total perimeter generally increases by about 10 units per additional hexagon beyond the first (which is 12).

Completing the Table

  • For 1 hexagon (x = 1):

    • Perimeter (y) = 12
    • Ordered Pair: (1, 12)

  • For 2 hexagons (x = 2):

    • Perimeter (y) = 22 (12 for the first + 10 for the second)
    • Ordered Pair: (2, 22)

  • For 3 hexagons (x = 3):

    • Perimeter (y) = 32 (12 for the first + 10 for the second + 10 for the third)
    • Ordered Pair: (3, 32)

Table

| Number of Hexagons (x) | Perimeter (y) | Ordered Pair (x, y) | |--------------------------|----------------|----------------------| | 1 | 12 | (1, 12) | | 2 | 22 | (2, 22) | | 3 | 32 | (3, 32) |

Part 2: Representing the relationship in words

The relationship between the number of hexagons and the perimeter of the figure they form shows a pattern where the perimeter starts at 12 for one hexagon, increasing by approximately 10 for each additional hexagon added. Therefore, the perimeter increases as more hexagons are added, but at a decreasing rate due to overlapping sides.

Part 3: Equation

The relationship can be represented with a linear equation: \[ y = 10x + 2 \] Where:

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

Part 4: Graph

To graph this relationship, plot the ordered pairs from the table:

  1. (1, 12)
  2. (2, 22)
  3. (3, 32)

On a coordinate plane:

  • x-axis represents the number of hexagons.
  • y-axis represents the perimeter.

You would see points that form a straight line, indicating a linear relationship.

Summary

The table, words, equation, and graph effectively summarize the relationship between the number of hexagons and their perimeter.