Rectangle Length Calculation: Area And Width Given

by Alex Johnson 51 views

Let's dive into the world of rectangles and explore how to calculate their length when we already know the area and width. This is a common problem in mathematics, particularly in algebra and geometry, and understanding the process can be incredibly useful. In this comprehensive guide, we'll break down the steps involved, explain the underlying concepts, and provide a clear, step-by-step method for solving such problems. So, if you've ever wondered how to find the missing side of a rectangle, you're in the right place!

Understanding the Basics of Rectangle Area

Before we jump into the calculation, let's quickly recap the basics of rectangle area. The area of a rectangle is the space enclosed within its four sides. It’s calculated by multiplying the length of the rectangle by its width. Mathematically, this relationship is expressed as:

Area = Length × Width

This simple formula is the cornerstone of our calculation. When you are given the area and one of the dimensions (either length or width), you can rearrange this formula to solve for the missing dimension. In our case, we are given the area and the width, and we need to find the length. To do this, we will rearrange the formula to isolate the length:

Length = Area / Width

This is the fundamental formula we'll use to solve our problem. Now that we have this basic understanding, let's tackle a specific example where the area and width are expressed as algebraic expressions.

Setting Up the Problem with Algebraic Expressions

Now, let's consider a scenario where the area of the rectangle is given by the algebraic expression (x^3 - 5x^2 + 3x - 15), and the width is given by (x^2 + 3). Our mission is to find the length, which will also be an algebraic expression. To do this, we’ll use the formula we derived earlier:

Length = Area / Width

In our case, this translates to:

Length = (x^3 - 5x^2 + 3x - 15) / (x^2 + 3)

This looks like a daunting division problem, but don't worry! We'll use a technique called polynomial long division to simplify this expression and find the length. Polynomial long division might seem a bit intimidating at first, but it's essentially the same process as regular long division, just with polynomials instead of numbers. It’s a powerful tool for simplifying rational expressions like the one we have here.

Performing Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another polynomial of a lower or equal degree. It helps us simplify complex expressions and find factors. Let's walk through the steps of performing polynomial long division with our given expressions:

  1. Set up the division: Write the dividend (x^3 - 5x^2 + 3x - 15) inside the division bracket and the divisor (x^2 + 3) outside the bracket.
  2. Divide the leading terms: Divide the first term of the dividend (x^3) by the first term of the divisor (x^2). This gives us x. Write x above the division bracket, aligned with the x term in the dividend.
  3. Multiply the quotient term by the divisor: Multiply the x we just found by the entire divisor (x^2 + 3). This gives us x^3 + 3x.
  4. Subtract and bring down: Subtract the result (x^3 + 3x) from the corresponding terms in the dividend. This leaves us with -5x^2 - 15. Bring down the next term from the dividend, which is -15, to get -5x^2 - 15.
  5. Repeat the process: Now, divide the first term of the new dividend (-5x^2) by the first term of the divisor (x^2). This gives us -5. Write -5 above the division bracket, aligned with the constant term in the dividend.
  6. Multiply and subtract again: Multiply -5 by the divisor (x^2 + 3), which gives us -5x^2 - 15. Subtract this from the current dividend (-5x^2 - 15). The result is 0, which means we have no remainder.

The result of the division is the quotient we found above the division bracket, which is x - 5. Therefore, the length of the rectangle is (x - 5).

Factoring as an Alternative Method

While polynomial long division is a reliable method, there's often a quicker way to solve these problems if you spot a pattern: factoring. Factoring involves breaking down the polynomial into simpler expressions that multiply together to give the original polynomial. In our case, the area expression (x^3 - 5x^2 + 3x - 15) can be factored by grouping.

Here’s how factoring by grouping works:

  1. Group terms: Group the first two terms and the last two terms together: (x^3 - 5x^2) + (3x - 15).
  2. Factor out common factors: Factor out the greatest common factor (GCF) from each group. From the first group, we can factor out x^2, and from the second group, we can factor out 3. This gives us x^2(x - 5) + 3(x - 5).
  3. Factor out the common binomial: Notice that both terms now have a common binomial factor of (x - 5). Factor this out to get (x - 5)(x^2 + 3).

So, the factored form of the area is (x - 5)(x^2 + 3). Now, remember our original equation:

Length = Area / Width

We know the area is (x - 5)(x^2 + 3) and the width is (x^2 + 3). Plugging these into the equation, we get:

Length = [(x - 5)(x^2 + 3)] / (x^2 + 3)

Now, we can see that the (x^2 + 3) terms cancel each other out, leaving us with:

Length = x - 5

This method confirms our result from the polynomial long division, but often in a more streamlined way. Factoring can be a valuable shortcut when it’s applicable, saving you time and effort.

Verifying the Solution

After finding the length, it's always a good practice to verify your solution. This helps ensure that you haven't made any mistakes along the way. To verify our solution, we can multiply the length we found (x - 5) by the given width (x^2 + 3) and see if it equals the given area (x^3 - 5x^2 + 3x - 15).

Let's multiply (x - 5) by (x^2 + 3):

(x - 5)(x^2 + 3) = x(x^2 + 3) - 5(x^2 + 3)

Distribute the x and -5:

= x^3 + 3x - 5x^2 - 15

Rearrange the terms to match the given area:

= x^3 - 5x^2 + 3x - 15

As you can see, the result matches the given area, which confirms that our solution for the length (x - 5) is correct. This verification step is a simple yet effective way to build confidence in your answer and catch any potential errors.

Practical Applications and Why This Matters

Understanding how to calculate the length of a rectangle given its area and width isn't just an academic exercise; it has practical applications in various real-world scenarios. Whether you're designing a room layout, calculating material needs for a construction project, or solving geometric problems in engineering, this skill comes in handy.

For example, imagine you're planning a garden and you know the total area you want your garden to cover, as well as the width of the space you have available. By using the formula we've discussed, you can easily determine the necessary length of the garden. Similarly, if you're working on an interior design project and need to fit a rug into a room, knowing the area of the rug and the width of the space can help you calculate the required length.

Beyond these practical applications, this type of problem also reinforces key mathematical concepts such as algebraic manipulation, polynomial division, and factoring. These are fundamental skills that are essential for success in higher-level mathematics and various STEM fields. By mastering these concepts, you're not just solving rectangle problems; you're building a solid foundation for tackling more complex mathematical challenges.

Conclusion: Mastering Rectangle Length Calculation

In conclusion, finding the length of a rectangle when you know its area and width is a straightforward process once you understand the underlying principles. By using the formula Length = Area / Width and employing techniques like polynomial long division or factoring, you can solve these problems with confidence. Remember to always verify your solution to ensure accuracy, and keep in mind the practical applications of this skill in various real-world scenarios.

By mastering this concept, you're not only expanding your mathematical toolkit but also enhancing your problem-solving abilities in general. Whether you're a student, a professional, or simply someone who enjoys mathematical puzzles, the ability to calculate rectangle dimensions is a valuable asset. So, keep practicing, keep exploring, and continue to build your mathematical prowess!

For further exploration of rectangle properties and related geometric concepts, visit Khan Academy's Geometry Section.