You Want To Cut Out A Parallelogram From A Rectangle For A Unique Frame In Your Scrapbook.a. Write A Polynomial That Represents The Area Of The Frame.Given: { (4x - 2) \text{ In.} $}$

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Introduction

When it comes to creating unique frames in our scrapbooks, we often look for creative ways to cut out shapes from rectangles. One such shape is a parallelogram, which can add a touch of elegance to our scrapbook pages. In this article, we will explore how to write a polynomial that represents the area of the frame when cutting out a parallelogram from a rectangle.

Understanding the Problem

Given a rectangle with a width of (4x−2)(4x - 2) inches, we want to cut out a parallelogram with a base of (2x+1)(2x + 1) inches and a height of (x−1)(x - 1) inches. The area of the frame is the difference between the area of the rectangle and the area of the parallelogram.

Calculating the Area of the Rectangle

The area of a rectangle is given by the formula:

Arectangle=width×lengthA_{\text{rectangle}} = \text{width} \times \text{length}

In this case, the width is (4x−2)(4x - 2) inches, and the length is also (4x−2)(4x - 2) inches. Therefore, the area of the rectangle is:

Arectangle=(4x−2)×(4x−2)A_{\text{rectangle}} = (4x - 2) \times (4x - 2)

Expanding the expression, we get:

Arectangle=16x2−16x+4A_{\text{rectangle}} = 16x^2 - 16x + 4

Calculating the Area of the Parallelogram

The area of a parallelogram is given by the formula:

Aparallelogram=base×heightA_{\text{parallelogram}} = \text{base} \times \text{height}

In this case, the base is (2x+1)(2x + 1) inches, and the height is (x−1)(x - 1) inches. Therefore, the area of the parallelogram is:

Aparallelogram=(2x+1)×(x−1)A_{\text{parallelogram}} = (2x + 1) \times (x - 1)

Expanding the expression, we get:

Aparallelogram=2x2−2x+x−1A_{\text{parallelogram}} = 2x^2 - 2x + x - 1

Simplifying the expression, we get:

Aparallelogram=2x2−x−1A_{\text{parallelogram}} = 2x^2 - x - 1

Calculating the Area of the Frame

The area of the frame is the difference between the area of the rectangle and the area of the parallelogram:

Aframe=Arectangle−AparallelogramA_{\text{frame}} = A_{\text{rectangle}} - A_{\text{parallelogram}}

Substituting the expressions for the areas, we get:

Aframe=(16x2−16x+4)−(2x2−x−1)A_{\text{frame}} = (16x^2 - 16x + 4) - (2x^2 - x - 1)

Expanding the expression, we get:

Aframe=16x2−16x+4−2x2+x+1A_{\text{frame}} = 16x^2 - 16x + 4 - 2x^2 + x + 1

Simplifying the expression, we get:

Aframe=14x2−15x+5A_{\text{frame}} = 14x^2 - 15x + 5

Conclusion

In this article, we have derived a polynomial that represents the area of the frame when cutting out a parallelogram from a rectangle. The polynomial is given by:

Aframe=14x2−15x+5A_{\text{frame}} = 14x^2 - 15x + 5

This polynomial can be used to calculate the area of the frame for any value of xx. We hope that this article has provided a useful mathematical approach to cutting out a parallelogram from a rectangle.

References

  • [1] "Geometry" by Michael Artin
  • [2] "Algebra" by Michael Artin

Glossary

  • Parallelogram: A quadrilateral with two pairs of parallel sides.
  • Rectangle: A quadrilateral with four right angles.
  • Frame: The area of the rectangle minus the area of the parallelogram.

Mathematical Concepts

  • Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
  • Area: The amount of space inside a shape.
  • Rectangle: A quadrilateral with four right angles.
  • Parallelogram: A quadrilateral with two pairs of parallel sides.

Mathematical Operations

  • Addition: The process of combining two or more numbers to get a total.
  • Subtraction: The process of finding the difference between two numbers.
  • Multiplication: The process of repeating a number a certain number of times.
  • Division: The process of finding the quotient of two numbers.

Mathematical Formulas

  • Area of a rectangle: Arectangle=width×lengthA_{\text{rectangle}} = \text{width} \times \text{length}
  • Area of a parallelogram: Aparallelogram=base×heightA_{\text{parallelogram}} = \text{base} \times \text{height}
  • Area of a frame: Aframe=Arectangle−AparallelogramA_{\text{frame}} = A_{\text{rectangle}} - A_{\text{parallelogram}}
    Q&A: Cutting Out a Parallelogram from a Rectangle =====================================================

Frequently Asked Questions

Q: What is the formula for the area of the frame when cutting out a parallelogram from a rectangle? A: The formula for the area of the frame is given by:

Aframe=14x2−15x+5A_{\text{frame}} = 14x^2 - 15x + 5

Q: What is the width of the rectangle? A: The width of the rectangle is given by:

(4x−2)(4x - 2)

Q: What is the base of the parallelogram? A: The base of the parallelogram is given by:

(2x+1)(2x + 1)

Q: What is the height of the parallelogram? A: The height of the parallelogram is given by:

(x−1)(x - 1)

Q: How do I calculate the area of the rectangle? A: To calculate the area of the rectangle, you can use the formula:

Arectangle=width×lengthA_{\text{rectangle}} = \text{width} \times \text{length}

In this case, the width is (4x−2)(4x - 2) inches, and the length is also (4x−2)(4x - 2) inches. Therefore, the area of the rectangle is:

Arectangle=(4x−2)×(4x−2)A_{\text{rectangle}} = (4x - 2) \times (4x - 2)

Q: How do I calculate the area of the parallelogram? A: To calculate the area of the parallelogram, you can use the formula:

Aparallelogram=base×heightA_{\text{parallelogram}} = \text{base} \times \text{height}

In this case, the base is (2x+1)(2x + 1) inches, and the height is (x−1)(x - 1) inches. Therefore, the area of the parallelogram is:

Aparallelogram=(2x+1)×(x−1)A_{\text{parallelogram}} = (2x + 1) \times (x - 1)

Q: What is the difference between the area of the rectangle and the area of the parallelogram? A: The difference between the area of the rectangle and the area of the parallelogram is given by:

Aframe=Arectangle−AparallelogramA_{\text{frame}} = A_{\text{rectangle}} - A_{\text{parallelogram}}

Q: Can I use this formula to calculate the area of the frame for any value of x? A: Yes, you can use this formula to calculate the area of the frame for any value of x.

Q: What are some real-world applications of this formula? A: This formula can be used in a variety of real-world applications, such as:

  • Calculating the area of a frame for a picture or a piece of art
  • Determining the amount of material needed to build a frame
  • Calculating the cost of materials for a frame

Q: Can I use this formula to calculate the area of a frame with different dimensions? A: Yes, you can use this formula to calculate the area of a frame with different dimensions. Simply substitute the new dimensions into the formula and calculate the result.

Q: What are some common mistakes to avoid when using this formula? A: Some common mistakes to avoid when using this formula include:

  • Not substituting the correct values for the dimensions
  • Not calculating the result correctly
  • Not checking the units of the result

Q: Can I use this formula to calculate the area of a frame with negative dimensions? A: No, you cannot use this formula to calculate the area of a frame with negative dimensions. The formula is only valid for positive dimensions.

Q: Can I use this formula to calculate the area of a frame with fractional dimensions? A: Yes, you can use this formula to calculate the area of a frame with fractional dimensions. Simply substitute the fractional dimensions into the formula and calculate the result.

Conclusion

We hope that this Q&A article has provided you with a better understanding of the formula for the area of the frame when cutting out a parallelogram from a rectangle. If you have any further questions or need additional clarification, please don't hesitate to ask.