Finding Rectangle Length: A Step-by-Step Guide

by Alex Johnson 47 views

Hey there, math enthusiasts! Today, we're diving into a geometry problem that's all about rectangles, area, and a bit of algebraic manipulation. Don't worry if the words "algebraic manipulation" sound intimidating. We'll break it down step by step, making it super easy to understand. So, grab your pencils, and let's unravel this intriguing problem together!

Understanding the Problem: The Area and Width of a Rectangle

Let's begin by clearly understanding what the problem presents to us. The core concept here revolves around the area of a rectangle. You know, that space enclosed within the four sides of a rectangle. We're given two crucial pieces of information:

  • The Area: The total area of the rectangle is represented by the expression x^3 - 5x^2 + 3x - 15.
  • The Width: The width of the rectangle is given as x^2 + 3.

Now, here's the tricky part. We need to find the length of this rectangle. Remember the fundamental formula that ties everything together? Area = Length × Width. Using this formula is the key to cracking this problem. Essentially, we have the area and the width, and we need to figure out what the length must be.

Breaking Down the Concepts: Area, Length, and Width

To make sure we're all on the same page, let's quickly recap what these terms mean in the context of a rectangle:

  • Area: The total space inside the rectangle. Think of it as the amount of carpet you'd need to cover the floor.
  • Length: The longer side of the rectangle. It's the distance from one end of the rectangle to the other.
  • Width: The shorter side of the rectangle. It's the distance from one side of the rectangle to the other.

The relationship is simple: multiply the length by the width, and you get the area. This is what we will use to find the length.

Solving for the Length: Step-by-Step Approach

Now that we've got a solid grasp of the problem, let's get into the step-by-step solution. We'll use a combination of algebraic techniques to find the length of the rectangle.

Step 1: Recall the Formula and Rearrange

First and foremost, let's write down the formula we're working with:

Area = Length × Width

Since we want to find the length, we need to rearrange this formula. To isolate the length, we can divide both sides of the equation by the width:

Length = Area / Width

This is the core of our solution. Now, we simply need to substitute the given expressions for the area and the width, and then simplify.

Step 2: Substitute the Given Values

Let's substitute the expressions we were given into the formula. Remember, the area is x^3 - 5x^2 + 3x - 15, and the width is x^2 + 3. So, our equation becomes:

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

Step 3: Factor the Expression for Area

Here’s where things get interesting. We will factor the expression in the numerator (the area) to simplify the division. We will factor by grouping:

  1. Group the terms: Group the first two terms and the last two terms together: (x^3 - 5x^2) + (3x - 15).
  2. Factor out the greatest common factor (GCF) from each group:
    • From the first group (x^3 - 5x^2), the GCF is x^2. Factoring it out, we get x^2(x - 5).
    • From the second group (3x - 15), the GCF is 3. Factoring it out, we get 3(x - 5).
  3. Rewrite the expression: Now our expression looks like this: x^2(x - 5) + 3(x - 5).
  4. Factor out the common binomial: Notice that both terms now have a common binomial factor of (x - 5). Factoring this out, we get (x - 5)(x^2 + 3).

So, the factored form of the area expression is (x - 5)(x^2 + 3).

Step 4: Simplify the Expression for Length

Now we can rewrite the equation for the length using the factored form of the area:

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

Notice that (x^2 + 3) appears in both the numerator and the denominator. We can cancel these out, which simplifies our equation dramatically.

Length = x - 5

Step 5: Identify the Answer

So, the length of the rectangle is x - 5. If you look back at the answer choices, you'll see that this matches option D. Congratulations, you've solved the problem!

Conclusion: Mastering Rectangle Area Problems

We did it! We successfully found the length of the rectangle. Through the process of understanding area, length, and width, and by using algebraic manipulation, we came to our solution. We learned how to rearrange formulas and factor expressions. This problem highlights how a solid grasp of basic algebraic principles can unlock solutions to geometric challenges.

Key Takeaways

  • Understanding the Formula: Always start by writing down the formula: Area = Length × Width.
  • Rearranging the Formula: If you need to find a different variable (like the length), rearrange the formula to isolate that variable.
  • Factoring: Factoring is a powerful tool for simplifying expressions and making it easier to solve for unknowns.
  • Step-by-Step Approach: Break down the problem into smaller, manageable steps. This will keep you organized and prevent mistakes.

Keep practicing, and you'll find yourself acing these types of problems in no time. You are now well-equipped to tackle similar problems with confidence! Keep exploring and enjoy the journey through math!

Further Exploration

If you're eager to learn more about geometry and algebra, here are some related topics you might enjoy:

  • Perimeter of Rectangles: Learn how to calculate the total distance around a rectangle.
  • Area of Other Shapes: Explore the formulas for the area of squares, triangles, circles, and more.
  • Factoring Techniques: Practice different factoring methods to strengthen your algebraic skills.
  • Polynomial Division: Dive deeper into dividing polynomials to solve more complex problems.

Keep up the great work, and don't hesitate to explore these related areas to enhance your math skills! Remember, practice makes perfect!

For more information, consider exploring resources at Khan Academy. This website provides comprehensive lessons and exercises on algebra and geometry, covering topics similar to the one discussed in this article, and many more. It's a great platform to strengthen your understanding and practice solving problems related to rectangle areas and other geometric concepts.