The Length Of A Rectangle Is 1 Cm More Than Its Width, And Its Area Is 28 Cm². Find Its Perimeter.

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Introduction

In mathematics, geometry plays a crucial role in understanding the properties of shapes and their dimensions. One of the fundamental concepts in geometry is the rectangle, which is a quadrilateral with four right angles. In this article, we will delve into the world of rectangles and explore a problem that involves finding the perimeter of a rectangle given its area and the relationship between its length and width.

Problem Statement

The problem states that the length of a rectangle is 1 cm more than its width, and its area is 28 cm². We need to find the perimeter of this rectangle.

Understanding the Relationship Between Length and Width

Let's denote the width of the rectangle as w cm. Since the length of the rectangle is 1 cm more than its width, we can express the length as (w + 1) cm.

Expressing the Area of the Rectangle

The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is 28 cm², so we can write the equation as:

28 = (w + 1) × w

Solving the Equation

To find the value of w, we need to solve the quadratic equation:

28 = w² + w

Rearranging the equation, we get:

w² + w - 28 = 0

Factoring the Quadratic Equation

We can factor the quadratic equation as:

(w + 7)(w - 4) = 0

Finding the Value of w

Setting each factor equal to zero, we get:

w + 7 = 0 or w - 4 = 0

Solving for w, we get:

w = -7 or w = 4

Discarding the Negative Value

Since the width of a rectangle cannot be negative, we discard the value w = -7. Therefore, the width of the rectangle is w = 4 cm.

Finding the Length of the Rectangle

Now that we have the value of w, we can find the length of the rectangle by substituting w into the expression (w + 1):

Length = (4 + 1) = 5 cm

Calculating the Perimeter of the Rectangle

The perimeter of a rectangle is given by the formula: Perimeter = 2 × (Length + Width). Substituting the values of length and width, we get:

Perimeter = 2 × (5 + 4) = 2 × 9 = 18 cm

Conclusion

In this article, we explored a problem that involved finding the perimeter of a rectangle given its area and the relationship between its length and width. We used algebraic techniques to solve the quadratic equation and found the value of the width. Finally, we calculated the perimeter of the rectangle using the formula for the perimeter of a rectangle.

Final Answer

The perimeter of the rectangle is 18 cm.

Related Topics

  • Geometry
  • Algebra
  • Quadratic Equations
  • Perimeter of a Rectangle

Further Reading

References

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Introduction

In our previous article, we explored a problem that involved finding the perimeter of a rectangle given its area and the relationship between its length and width. We used algebraic techniques to solve the quadratic equation and found the value of the width. In this article, we will answer some frequently asked questions related to the problem.

Q&A Session

Q: What is the relationship between the length and width of the rectangle?

A: The length of the rectangle is 1 cm more than its width. If we denote the width as w cm, then the length can be expressed as (w + 1) cm.

Q: How do we find the area of the rectangle?

A: The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is 28 cm².

Q: What is the quadratic equation that we need to solve?

A: The quadratic equation is w² + w - 28 = 0.

Q: How do we factor the quadratic equation?

A: We can factor the quadratic equation as (w + 7)(w - 4) = 0.

Q: What are the possible values of w?

A: The possible values of w are w = -7 and w = 4. However, since the width of a rectangle cannot be negative, we discard the value w = -7.

Q: What is the length of the rectangle?

A: The length of the rectangle is (w + 1) = (4 + 1) = 5 cm.

Q: How do we calculate the perimeter of the rectangle?

A: The perimeter of a rectangle is given by the formula: Perimeter = 2 × (Length + Width). Substituting the values of length and width, we get Perimeter = 2 × (5 + 4) = 2 × 9 = 18 cm.

Q: What is the final answer?

A: The perimeter of the rectangle is 18 cm.

Related Questions

  • What is the formula for the area of a rectangle?
  • How do we find the length of a rectangle given its width and the relationship between its length and width?
  • What is the formula for the perimeter of a rectangle?
  • How do we solve a quadratic equation?

Answers

  • The formula for the area of a rectangle is Area = Length × Width.
  • To find the length of a rectangle given its width and the relationship between its length and width, we can use the expression (w + 1), where w is the width.
  • The formula for the perimeter of a rectangle is Perimeter = 2 × (Length + Width).
  • To solve a quadratic equation, we can use algebraic techniques such as factoring or the quadratic formula.

Further Reading

References