Question

on your paper, write expressions for the area and perimeter of the shape shown at right. then calculate the area and perimeter of the shape for each x - value.

Explanation:

To solve this, we analyze the shape (assuming it's made of rectangles/squares with side \( x \)):

Step 1: Area Calculation

Count the number of unit squares (each with area \( x^2 \)):

• Top 3 rectangles: \( 3 \times 3x \)? Wait, no—looking at the diagram (3 tall rectangles, 2 smaller below). Wait, let's re-express:
• 3 vertical rectangles (height \( 3x \), width \( x \))? No, better: count all squares/rectangles. Let's assume each small square has side \( x \).
• Top row: 3 rectangles (each \( x \times 3x \)? No, maybe:

Wait, the diagram shows: 3 long rectangles (each \( x \times 3x \)? No, let's count the number of \( x \times x \) squares:

• 3 long rectangles: each is \( x \times 3x \)? No, maybe:

Wait, the correct way:

• Area: Sum of areas of all parts. Let's say:
• 3 rectangles of size \( x \times 3x \)? No, maybe the shape has:
• 3 tall rectangles (height \( 3x \), width \( x \)): area \( 3 \times (x \times 3x) = 9x^2 \)? No, that's wrong. Wait, the diagram (from the image) likely has:
• Top 3 rectangles (each \( x \times 3x \))? No, let's look at the bottom: 2 small squares and 2 small rectangles? Wait, maybe the total number of \( x \times x \) squares is \( 3 \times 3 + 2 + 1 = 12 \)? No, this is unclear. Wait, the problem says "the shape shown"—assuming it's a composite of rectangles with side \( x \), let's re-express:

Correct Approach (Assuming the Shape is Made of \( x \times x \) Squares):

• Let’s count the number of \( x \times x \) squares:
• Top 3 rectangles: each is \( x \times 3x \)? No, maybe the shape is:
• 3 vertical rectangles (height \( 3x \), width \( x \)): area \( 3 \times (x \times 3x) = 9x^2 \)? No, that's not right. Wait, the bottom has 2 small squares and 2 small rectangles? Wait, maybe the area is \( 3 \times 3x \times x + 2 \times x \times x + 1 \times x \times x = 9x^2 + 2x^2 + x^2 = 12x^2 \)? No, this is confusing.

Step 2: Perimeter Calculation

Perimeter: Sum of all outer sides. Let's assume the shape has a total length and width. If the shape is symmetric, the perimeter can be calculated by finding the total outer length.

Alternative (Assuming the Shape is a Rectangle with Adjustments):

Wait, the problem is incomplete (no \( x \)-value given, and the diagram is unclear). But to proceed, let's assume \( x = 1 \) (for example) or re-express:

If \( x = 1 \):

• Area: Count the number of squares. Let's say the shape has 12 squares (each \( 1 \times 1 \)), so area \( 12 \).
• Perimeter: Sum of outer sides. If the shape is, say, 5 units long and 4 units wide (adjusting for the indent), perimeter \( 2 \times (5 + 4) = 18 \) (if \( x = 1 \)).

But since the problem is unclear (no \( x \)-value or exact diagram), we need more info. However, assuming \( x = 2 \) (example):

Example with \( x = 2 \):

• Area: If there are 12 \( x \times x \) squares, area \( 12 \times (2)^2 = 48 \).
• Perimeter: If the outer length is \( 5x \) and width \( 4x \), perimeter \( 2(5x + 4x) = 18x \). For \( x = 2 \), perimeter \( 36 \).

Final Answer (Assuming \( x = 2 \) and 12 squares):

• Area: \( 12x^2 = 12(2)^2 = 48 \)
• Perimeter: \( 18x = 18(2) = 36 \)

(Note: The exact answer depends on the precise shape, which is unclear from the image. The above is a general approach.)

Answer:

To solve this, we analyze the shape (assuming it's made of rectangles/squares with side \( x \)):

Step 1: Area Calculation

Count the number of unit squares (each with area \( x^2 \)):

• Top 3 rectangles: \( 3 \times 3x \)? Wait, no—looking at the diagram (3 tall rectangles, 2 smaller below). Wait, let's re-express:
• 3 vertical rectangles (height \( 3x \), width \( x \))? No, better: count all squares/rectangles. Let's assume each small square has side \( x \).
• Top row: 3 rectangles (each \( x \times 3x \)? No, maybe:

Wait, the diagram shows: 3 long rectangles (each \( x \times 3x \)? No, let's count the number of \( x \times x \) squares:

• 3 long rectangles: each is \( x \times 3x \)? No, maybe:

Wait, the correct way:

• Area: Sum of areas of all parts. Let's say:
• 3 rectangles of size \( x \times 3x \)? No, maybe the shape has:
• 3 tall rectangles (height \( 3x \), width \( x \)): area \( 3 \times (x \times 3x) = 9x^2 \)? No, that's wrong. Wait, the diagram (from the image) likely has:
• Top 3 rectangles (each \( x \times 3x \))? No, let's look at the bottom: 2 small squares and 2 small rectangles? Wait, maybe the total number of \( x \times x \) squares is \( 3 \times 3 + 2 + 1 = 12 \)? No, this is unclear. Wait, the problem says "the shape shown"—assuming it's a composite of rectangles with side \( x \), let's re-express:

Correct Approach (Assuming the Shape is Made of \( x \times x \) Squares):

• Let’s count the number of \( x \times x \) squares:
• Top 3 rectangles: each is \( x \times 3x \)? No, maybe the shape is:
• 3 vertical rectangles (height \( 3x \), width \( x \)): area \( 3 \times (x \times 3x) = 9x^2 \)? No, that's not right. Wait, the bottom has 2 small squares and 2 small rectangles? Wait, maybe the area is \( 3 \times 3x \times x + 2 \times x \times x + 1 \times x \times x = 9x^2 + 2x^2 + x^2 = 12x^2 \)? No, this is confusing.

Step 2: Perimeter Calculation

Perimeter: Sum of all outer sides. Let's assume the shape has a total length and width. If the shape is symmetric, the perimeter can be calculated by finding the total outer length.

Alternative (Assuming the Shape is a Rectangle with Adjustments):

Wait, the problem is incomplete (no \( x \)-value given, and the diagram is unclear). But to proceed, let's assume \( x = 1 \) (for example) or re-express:

If \( x = 1 \):

• Area: Count the number of squares. Let's say the shape has 12 squares (each \( 1 \times 1 \)), so area \( 12 \).
• Perimeter: Sum of outer sides. If the shape is, say, 5 units long and 4 units wide (adjusting for the indent), perimeter \( 2 \times (5 + 4) = 18 \) (if \( x = 1 \)).

But since the problem is unclear (no \( x \)-value or exact diagram), we need more info. However, assuming \( x = 2 \) (example):

Example with \( x = 2 \):

• Area: If there are 12 \( x \times x \) squares, area \( 12 \times (2)^2 = 48 \).
• Perimeter: If the outer length is \( 5x \) and width \( 4x \), perimeter \( 2(5x + 4x) = 18x \). For \( x = 2 \), perimeter \( 36 \).

Final Answer (Assuming \( x = 2 \) and 12 squares):

• Area: \( 12x^2 = 12(2)^2 = 48 \)
• Perimeter: \( 18x = 18(2) = 36 \)

(Note: The exact answer depends on the precise shape, which is unclear from the image. The above is a general approach.)