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

Question

1
3
1
1
1
1
1
3
3
1 rectangle
3 rectangles
2 rectangles
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Rectangles​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
  
8
  
left parenthesis 1 comma 8 right parenthesis
2
  
10
  
left parenthesis 2 comma 10 right parenthesis
3
  
12
  
left parenthesis 3 comma 12 right parenthesis
​(Simplify your​ answers.)
Part 2
Represent the relationship using words.
Multiply the number of rectangles in each figure by
  
enter your response here​, to get the total length of the
▼
left and right
top and bottom
sides of the combined figure. Then add
  
enter your response here for the total length of the
▼
left and right
top and bottom
sides of the combined figure to get the entire perimeter.

Answer 1

Let's break down the relationship between the number of rectangles (x) and the perimeter of the figure they form (y).

Part 1: Table Representation

We need to fill out the table based on the provided information. The perimeter shown for each number of rectangles implies a pattern. Here’s how it can look:

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

Part 2: Words Representation

Multiply the number of rectangles in each figure by 2 to get the total length of the left and right sides of the combined figure. Then add 4 for the total length of the top and bottom sides of the combined figure to get the entire perimeter.

Observations for the Relationship

• For every additional rectangle added, the perimeter increases by 2.
• The relationship between the number of rectangles (x) and the perimeter (y) can be expressed mathematically.

Part 3: Equation Representation

From the patterns seen in our table, we can formulate the equation:

\[ y = 2x + 6 \]

Where:

• \( y \) is the perimeter,
• \( x \) is the number of rectangles,
• The constant \( 6 \) comes from the initial perimeter when there is 1 rectangle (the base length).

Part 4: Graph Representation

To graph the relationship, create a Cartesian coordinate system where the x-axis represents the number of rectangles and the y-axis represents the perimeter.

1. Plot the points from the ordered pairs: (1, 8), (2, 10), (3, 12)
2. The points will create a linear relationship which can be connected with a straight line.
3. Label the axes and draw the line to show the increasing perimeter with each additional rectangle.

Conclusion: This illustrates a linear relationship where the perimeter increases predictably with the number of rectangles added to the figure. The equation \( y = 2x + 6 \) encapsulates this relationship, and the graph visually depicts it.