Composite Rectangle Area Algebra

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Composite rectangle with algebraic expressions

📐 Part 1: Finding Areas with Algebra

1. A composite rectangle has dimensions (x + 3) by (2x + 5). Which expression represents its area?

2x² + 11x + 15

3x + 8

2x² + 5x + 6x + 15

x² + 8x + 15

2. A rectangular garden is divided into two sections. The first section is 4 feet by (x + 2) feet, and the second section is 6 feet by (x + 2) feet. What is the total area?

10x + 20

(x + 2)(10)

4x + 8 + 6x + 12

All of the above

3. Which property is used when rewriting 3(2x + 4) as 6x + 12?

Commutative Property

Associative Property

Distributive Property

Identity Property

4. A composite rectangle can be split into two rectangles with areas 5(x + 3) and 2(x + 3). What is the total area in simplified form?

✏️ Part 2: Problem Solving with Composite Rectangles

5. A parking lot is designed as a composite rectangle. The main section is (3x + 1) feet by (x + 4) feet. An additional section is (2x - 1) feet by (x + 4) feet.

a) Write an expression for the total area using the distributive property:

b) Expand and simplify your expression:

c) If x = 8 feet, what is the total area of the parking lot?

6. A rectangular poster is 15 inches wide and (2y + 7) inches tall. It has a rectangular border that adds 3 inches to each side. Write and simplify an expression for the total area of the poster including the border.

7. Explain how the distributive property helps you find the area of composite rectangles that share a common dimension. Give an example.

🎯 Part 3: Challenge Problems

8. A composite rectangle has a total area of 6x² + 21x + 18. If one dimension is (2x + 3), what is the other dimension?

9. Two rectangular flower beds share a common width of (w + 5) feet. The first bed has length 8 feet and the second has length (2w - 3) feet. If the total area is 135 square feet, find the value of w.