Area Of A Rectangle Explained Step-by-Step Solution And Examples

by Kenji Nakamura 65 views

Hey guys! Today, we're diving into a fundamental concept in geometry: calculating the area of a rectangle. This isn't just some abstract math problem; understanding area is super practical, whether you're figuring out how much carpet you need for a room or planning a garden layout. We'll break down a specific problem, but more importantly, we'll build a solid understanding of the underlying principles. So, let's get started!

The Area of a Rectangle The Basics

Before we tackle our specific question, let's quickly review the basics of finding the area of a rectangle. The area, as you probably remember, is the amount of two-dimensional space a shape covers. For a rectangle, it's a pretty straightforward calculation:

Area = Length × Width

That's it! The area of a rectangle is simply the product of its length and width. Make sure both dimensions are in the same units (e.g., both in inches, both in meters) before you multiply. Now that we've refreshed the foundation, let's move on to the problem at hand.

Decoding the Problem A Step-by-Step Approach

Okay, here's the core of our challenge: Which of the following represents the area of a rectangle whose length is 3x + 5 and whose width is x - 2?

We've got four options to choose from:

A. 3x^2 - x - 10 B. 3x^2 - 10 C. 3x^2 + x - 10 D. 3x^2 - 11x - 10

Don't let the algebraic expressions intimidate you. We're just going to apply the same basic formula, but with a little algebra mixed in. Here's how we'll break it down:

  1. Write down the formula: Area = Length × Width
  2. Substitute the given expressions: Area = (3x + 5) × (x - 2)
  3. Expand the expression: This is where the distributive property (or the FOIL method) comes in handy. We'll multiply each term in the first parenthesis by each term in the second.
  4. Simplify the expression: Combine any like terms to get our final answer.

Let's walk through the expansion and simplification steps in detail.

Expanding the Expression Unleashing the Distributive Property

The key to solving this problem is correctly expanding the expression (3x + 5) × (x - 2). We'll use the distributive property, which states that a(b + c) = ab + ac. You might also know this as the FOIL method (First, Outer, Inner, Last), which is just a mnemonic for applying the distributive property in a specific order. Here's how it works:

  • First: Multiply the first terms in each parenthesis: 3x * x = 3x^2
  • Outer: Multiply the outer terms: 3x * -2 = -6x
  • Inner: Multiply the inner terms: 5 * x = 5x
  • Last: Multiply the last terms: 5 * -2 = -10

Now, we add all these terms together: 3x^2 - 6x + 5x - 10

Simplifying the Expression Combining Like Terms

We're almost there! Our expanded expression is 3x^2 - 6x + 5x - 10. To simplify, we need to combine the "like terms." Like terms are those that have the same variable raised to the same power. In this case, -6x and 5x are like terms. Combining them is simple: -6x + 5x = -x

So, our simplified expression becomes: 3x^2 - x - 10

The Solution Unveiled Choosing the Correct Answer

Comparing our simplified expression, 3x^2 - x - 10, to the answer choices, we see that it matches option A. Therefore, the area of the rectangle is represented by the expression 3x^2 - x - 10.

Why the Other Options Are Incorrect Spotting Common Mistakes

It's just as important to understand why the other options are wrong as it is to know why the correct answer is right. This helps you avoid making similar mistakes in the future. Let's take a quick look at the other options:

  • B. 3x^2 - 10: This option is missing the -x term. This likely results from forgetting to multiply the inner and outer terms when expanding the expression.
  • C. 3x^2 + x - 10: This option has a +x term instead of -x. This probably comes from incorrectly combining the -6x and 5x terms, perhaps adding them instead of subtracting.
  • D. 3x^2 - 11x - 10: This option has -11x instead of -x. This suggests an error in combining the -6x and 5x terms, possibly adding them as if they were both negative.

By understanding these common pitfalls, you can be more careful when expanding and simplifying algebraic expressions.

Level Up Your Skills Practice Makes Perfect

To really master this concept, practice is key! Try working through similar problems with different lengths and widths. You can even create your own problems to challenge yourself. Here are a few ideas to get you started:

  1. Change the expressions: Instead of 3x + 5 and x - 2, try something like 2x - 1 and x + 4.
  2. Increase the complexity: Use expressions with more terms, such as x^2 + 2x - 3 as a side length.
  3. Work backwards: Give yourself an area expression and one side length, and try to find the other side length.

Real-World Applications Where Area Calculations Shine

The area of a rectangle might seem like a purely mathematical concept, but it has tons of real-world applications. Here are just a few examples:

  • Home improvement: Figuring out how much flooring, paint, or wallpaper you need for a room.
  • Gardening: Calculating the area of a rectangle of a garden bed to determine how much soil or fertilizer to use.
  • Construction: Determining the amount of materials needed for building walls, roofs, or other structures.
  • Design: Laying out rooms in a building, designing furniture, or creating graphics.

Understanding area of a rectangle calculations is a fundamental skill that can save you time, money, and frustration in many different situations.

The Takeaway Mastering the Area of a Rectangle

We've covered a lot in this guide, from the basic formula for the area of a rectangle to expanding and simplifying algebraic expressions. The key takeaways are:

  • The area of a rectangle is found by multiplying its length and width.
  • When dealing with algebraic expressions, use the distributive property (or FOIL method) to expand.
  • Combine like terms to simplify expressions.
  • Practice is crucial for mastering these concepts.
  • Understanding area has numerous real-world applications.

So, the next time you encounter a problem involving the area of a rectangle, you'll be well-equipped to tackle it with confidence. Keep practicing, and you'll be a geometry pro in no time! Remember, math isn't just about numbers and formulas; it's about developing problem-solving skills that you can use in all aspects of life. And most importantly, have fun with it!

Which expression correctly represents the area of a rectangle with a length of 3x + 5 and a width of x - 2?

Area of a Rectangle Explained Step-by-Step Solution and Examples