Calculating The Length Of A Rectangular Window With Area 2646 Cm²

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In this article, we'll dive into a practical math problem involving calculating the length of a rectangular window. This type of problem often appears in standardized tests and real-world applications, so understanding the steps to solve it is essential. We'll break down the problem, explore the concepts involved, and provide a clear solution. This article will help you understand how to solve area problems involving rectangles, particularly when there's a relationship between the length and width.

Problem Statement

We have a rectangular window with an area of 2646 cm². The length of the window is 50% greater than its width. The question is: what is the length of the window?

Breaking Down the Problem

To solve this problem, we'll need to use our knowledge of the area of a rectangle and some basic algebra. Here’s a breakdown of the steps we’ll take:

  1. Define variables: We'll represent the unknown dimensions (length and width) with variables.
  2. Express the relationship between length and width: We'll translate the statement “the length is 50% greater than the width” into an algebraic equation.
  3. Use the area formula: We'll use the formula for the area of a rectangle (Area = length × width) to set up an equation.
  4. Solve the equation: We'll solve the equation to find the width of the window.
  5. Calculate the length: Once we have the width, we'll use the relationship between length and width to find the length.

Solving the Problem Step-by-Step

Now, let's walk through the solution step-by-step:

1. Define Variables

Let:

  • w represent the width of the window in centimeters.
  • l represent the length of the window in centimeters.

2. Express the Relationship Between Length and Width

The problem states that the length is 50% greater than the width. This means the length is equal to the width plus 50% of the width. We can write this as:

l = w + 0.50w

Simplifying this equation, we get:

l = 1.5w

This equation tells us that the length is 1.5 times the width.

3. Use the Area Formula

The area of a rectangle is given by the formula:

Area = length × width

We know the area is 2646 cm², so we can write the equation:

2646 = l × w

Now, we can substitute the expression for l from step 2 (l = 1.5w) into this equation:

2646 = (1.5w) × w

4. Solve the Equation

Simplifying the equation, we get:

2646 = 1.5w²

To solve for w², we divide both sides of the equation by 1.5:

w² = 2646 / 1.5

w² = 1764

Now, we take the square root of both sides to find w:

w = √1764

w = 42

So, the width of the window is 42 cm.

5. Calculate the Length

We know that l = 1.5w, so we can substitute the value of w we just found:

l = 1.5 × 42

l = 63

Therefore, the length of the window is 63 cm.

Final Answer

The length of the rectangular window is 63 cm.

Key Concepts

This problem highlights several key mathematical concepts:

  • Area of a rectangle: Understanding the formula Area = length × width is crucial for solving problems involving rectangular shapes.
  • Percentage increase: Knowing how to express a percentage increase algebraically (in this case, length being 50% greater than width) is essential.
  • Algebraic equations: Setting up and solving algebraic equations is a fundamental skill in mathematics. This problem requires substituting one equation into another and solving for an unknown variable.
  • Square roots: Calculating square roots is necessary to find the value of a variable when it's squared.

Mastering these concepts will not only help you solve similar problems but also build a strong foundation in mathematics.

Why This Matters

Understanding how to calculate dimensions and areas is more than just a math exercise. It has practical applications in various fields, including:

  • Construction and architecture: Calculating the dimensions of rooms, windows, and other structures is essential for building design and construction.
  • Interior design: Knowing how to calculate areas helps in planning furniture layouts and determining the amount of materials needed, like flooring or wallpaper.
  • Real estate: Understanding property sizes and dimensions is important for buying, selling, and managing real estate.
  • Everyday life: From gardening (calculating the area of a garden bed) to DIY projects, knowing how to work with dimensions and areas is a valuable skill.

Practice Problems

To solidify your understanding, try solving these similar problems:

  1. A rectangular garden has an area of 48 square meters. The length is twice the width. Find the length and width of the garden.
  2. A rectangular painting has an area of 1200 square inches. The width is 25% less than the length. What are the dimensions of the painting?
  3. The area of a rectangular floor is 360 square feet. If the length is 24 feet, what is the width?

Working through these problems will help you develop your problem-solving skills and gain confidence in applying the concepts we've discussed.

Conclusion

Calculating the dimensions of a rectangle, especially when there's a relationship between the length and width, involves a combination of geometry and algebra. By breaking down the problem into steps, defining variables, and using the area formula, we can successfully find the solution. This problem demonstrates the practical application of mathematical concepts in real-world scenarios. So, embrace the challenge, practice your skills, and you'll be well-equipped to tackle similar problems in the future.

This article has explored a specific problem, but the principles discussed apply broadly to problems involving areas and dimensions. Remember to always read the problem carefully, identify the key information, and use the appropriate formulas and techniques to find the solution. The key to success in math is practice, so keep working at it, and you'll see improvement over time.