Rectangle Width: Area & Length Calculation Guide

Hey guys! Let's dive into a common math problem: finding the width of a rectangle when you know its area and length. This is a super practical skill, whether you're tackling homework or even planning a home improvement project. We'll break down the steps using the example where the area (${A}$) of a rectangle is given by ${120x^2 + 78x - 90}$ and the length (${l}$) is ${12x + 15}$. Ready to get started?

Understanding the Basics

Before we jump into the calculations, let’s make sure we're all on the same page with the fundamentals. The area of a rectangle is found by multiplying its length and width. Mathematically, this is represented as:

${ A = l \times w }$

Where:

• ${ A }$ is the area of the rectangle.
• ${ l }$ is the length of the rectangle.
• ${ w }$ is the width of the rectangle.

In our problem, we know the area ${A = 120x^2 + 78x - 90}$ and the length ${l = 12x + 15}$. Our mission is to find the width ${w}$.

To do this, we need to rearrange the formula to solve for ${w}$. We can divide both sides of the equation by the length ${l}$:

${ w = \frac{A}{l} }$

Now, we can substitute the given expressions for ${A}$ and ${l}$ into this formula. Let's get into the nitty-gritty of the calculation!

Step-by-Step Calculation

Okay, let's get those math muscles working! We're going to find the width (${w}$) by dividing the area (${A}$) by the length (${l}$). Here’s how we set it up:

${ w = \frac{120x^2 + 78x - 90}{12x + 15} }$

This looks a bit intimidating, but don't worry, we'll tackle it step by step. The first thing we want to do is see if we can simplify the expression by factoring. Factoring helps us break down complex expressions into simpler terms, which makes division much easier.

Factoring the Area

Let's factor the area expression ${120x^2 + 78x - 90}$. First, we look for the greatest common factor (GCF) of the coefficients. The GCF of 120, 78, and -90 is 6. Factoring out 6, we get:

${ 120x^2 + 78x - 90 = 6(20x^2 + 13x - 15) }$

Now, we need to factor the quadratic expression ${20x^2 + 13x - 15}$. We're looking for two binomials that multiply to give us this quadratic. This can be done by splitting the middle term or using the quadratic formula, but in this case, let’s try factoring by grouping. We need two numbers that multiply to ${20 \times -15 = -300}$ and add up to 13. Those numbers are 25 and -12. So, we rewrite the middle term:

${ 20x^2 + 13x - 15 = 20x^2 + 25x - 12x - 15 }$

Now, we factor by grouping:

${ (20x^2 + 25x) + (-12x - 15) = 5x(4x + 5) - 3(4x + 5) }$

We see a common factor of ${(4x + 5)}$, so we factor that out:

${ 5x(4x + 5) - 3(4x + 5) = (5x - 3)(4x + 5) }$

Putting it all together, the factored form of the area is:

${ 120x^2 + 78x - 90 = 6(5x - 3)(4x + 5) }$

Factoring the Length

Next, let's factor the length expression ${12x + 15}$. The GCF of 12 and 15 is 3, so we factor that out:

${ 12x + 15 = 3(4x + 5) }$

Dividing the Factored Expressions

Now we have the factored forms of both the area and the length. We can rewrite the expression for the width as:

${ w = \frac{6(5x - 3)(4x + 5)}{3(4x + 5)} }$

We can now cancel common factors. Notice that we have a ${(4x + 5)}$ in both the numerator and the denominator, so we can cancel those out. Also, we can simplify the fraction ${\frac{6}{3}}$ to 2:

${ w = \frac{6(5x - 3)(4x + 5)}{3(4x + 5)} = 2(5x - 3) }$

Simplifying the Result

Finally, we distribute the 2 to the terms inside the parentheses:

${ w = 2(5x - 3) = 10x - 6 }$

So, the width of the rectangle is ${10x - 6}$. Awesome job! We've successfully navigated through the factoring and division. Let’s recap what we’ve done to make sure we’ve got it down pat.

Reviewing the Solution

Let's take a moment to recap how we found the width of the rectangle. Here’s a quick run-through of the steps:

1. Start with the Formula: We knew the area (${A}$) and length (${l}$) and needed to find the width (${w}$). We used the formula ${A = l \times w}$ and rearranged it to ${w = \frac{A}{l}}$.

2. Substitute the Expressions: We plugged in the given expressions for the area and length:

   ${ w = \frac{120x^2 + 78x - 90}{12x + 15} }$

3. Factor the Area: We factored the area expression ${120x^2 + 78x - 90}$ by first factoring out the GCF of 6, and then factoring the resulting quadratic expression. This gave us:

   ${ 120x^2 + 78x - 90 = 6(5x - 3)(4x + 5) }$

4. Factor the Length: We factored the length expression ${12x + 15}$ by factoring out the GCF of 3, which gave us:

   ${ 12x + 15 = 3(4x + 5) }$

5. Divide the Factored Expressions: We rewrote the division problem with the factored expressions:

   ${ w = \frac{6(5x - 3)(4x + 5)}{3(4x + 5)} }$

6. Cancel Common Factors: We canceled the common factor of ${(4x + 5)}$ and simplified the fraction ${\frac{6}{3}}$ to 2:

   ${ w = 2(5x - 3) }$

7. Simplify: Finally, we distributed the 2 to get the width:

   ${ w = 10x - 6 }$

So, the width of the rectangle is ${10x - 6}$. We found the answer by carefully factoring and simplifying the expressions. Factoring is a super handy tool in algebra, and this problem really shows how useful it can be. By breaking down the expressions into smaller parts, we made the division much easier to manage.

Common Mistakes to Avoid

Alright, let's chat about some common pitfalls that students often encounter when tackling problems like this. Knowing these can help you steer clear of errors and boost your confidence. Here are a few key things to watch out for:

1. Forgetting to Factor: This is a big one! Jumping straight into division without factoring can make the problem way more complicated than it needs to be. Always look for opportunities to factor first, as it simplifies the expressions and makes cancellations possible. In our example, factoring both the area and the length was crucial to solving the problem efficiently.
2. Incorrect Factoring: Factoring mistakes can throw off your entire solution. Double-check your factored expressions by multiplying them back together to ensure they match the original expression. For instance, make sure that ${6(5x - 3)(4x + 5)}$ really does equal ${120x^2 + 78x - 90}$. A quick check can save you a lot of trouble!
3. Missing the GCF: Always look for the greatest common factor (GCF) before attempting more complex factoring techniques. Factoring out the GCF first makes the remaining expression simpler and easier to work with. In our problem, we factored out 6 from the area expression and 3 from the length expression. Missing this step can lead to more complicated factoring later on.
4. Canceling Terms Incorrectly: You can only cancel factors, not terms. Factors are expressions that are multiplied together, while terms are added or subtracted. For example, you can cancel ${(4x + 5)}$ because it’s a factor in both the numerator and the denominator, but you can’t cancel individual terms within the parentheses. Remember, canceling terms incorrectly can lead to a wrong answer.
5. Sign Errors: Pay close attention to signs, especially when factoring and distributing. A simple sign error can change the whole problem. For example, when factoring ${20x^2 + 13x - 15}$, getting the signs right is crucial. Double-check that your factored expression expands back to the original expression with the correct signs.
6. Skipping Steps: It might be tempting to skip steps to save time, but this can lead to mistakes. Write out each step clearly, especially when factoring and simplifying. This way, you can easily review your work and catch any errors.

By keeping these common mistakes in mind, you’ll be better equipped to tackle similar problems with confidence and accuracy. Remember, math is all about practice, so the more you work through these types of problems, the better you’ll become!

Real-World Applications

Okay, so we've nailed the math, but let's think about why this stuff matters in the real world. Finding the width of a rectangle when you know its area and length isn't just a textbook problem; it's something that comes up in various practical situations. Let’s explore a few examples.

1. Home Improvement Projects: Imagine you’re planning to install new flooring in a rectangular room. You know the total area you want to cover and the length of the room. To figure out how much material you need, you'll need to calculate the width. This is exactly the kind of problem we've been solving! Whether it's tiling a bathroom, carpeting a living room, or laying hardwood floors, understanding how to calculate dimensions is essential for accurate planning and budgeting.
2. Gardening and Landscaping: Gardeners often need to calculate areas and dimensions when planning garden beds or landscaping projects. If you know the total area you want for a garden and the length of one side, you can use the formula we’ve discussed to find the width. This helps in determining how much soil, mulch, or fencing you'll need. Accurate measurements ensure you don’t overspend or run short on materials.
3. Construction and Architecture: In construction, calculating dimensions is a daily task. Architects and builders need to determine the sizes of rooms, the amount of materials needed, and the layout of structures. Knowing how to find the width of a rectangular space given its area and length is a fundamental skill for anyone in these fields. It’s crucial for ensuring that buildings are constructed according to plan and that materials are used efficiently.
4. Interior Design: Interior designers frequently work with room dimensions to plan furniture layouts and create functional spaces. If a designer knows the area of a room and one dimension (say, the length), they can calculate the other dimension (the width) to make sure furniture fits properly and the space is well-utilized. This ensures that the design is both aesthetically pleasing and practical.
5. Real Estate: When buying, selling, or renting property, knowing the dimensions of a space is vital. Real estate agents and potential buyers often calculate areas and dimensions to assess the size and layout of a property. Understanding how to find the width of a room can help buyers determine if their furniture will fit or if the space meets their needs. For sellers, accurate measurements can help in marketing the property effectively.

These are just a few examples, but they highlight how useful this mathematical skill is in everyday life. By mastering these calculations, you’re not just acing your math tests; you’re also equipping yourself with valuable skills for practical applications. So, keep practicing and remember that math is a tool that helps us understand and interact with the world around us!

Practice Problems

To really nail this concept, practice makes perfect! Let's try a couple more problems to solidify your understanding. Grab a pen and paper, and let’s work through these together.

Problem 1

The area of a rectangle is given by ${A = 6x^2 + 11x - 10}$, and the length is ${l = 2x + 5}$. Find the width ${w}$.

Solution

1. Write the formula: ${w = \frac{A}{l}}$

2. Substitute:

   ${ w = \frac{6x^2 + 11x - 10}{2x + 5} }$

3. Factor the area: We need to factor ${6x^2 + 11x - 10}$. Look for two numbers that multiply to ${6 \times -10 = -60}$ and add up to 11. Those numbers are 15 and -4. So, we rewrite the middle term:

   ${ 6x^2 + 11x - 10 = 6x^2 + 15x - 4x - 10 }$

   Factor by grouping:

   ${ (6x^2 + 15x) + (-4x - 10) = 3x(2x + 5) - 2(2x + 5) }$

   Factor out the common term ${(2x + 5)}$:

   ${ 3x(2x + 5) - 2(2x + 5) = (3x - 2)(2x + 5) }$

4. Factor the length: The length ${2x + 5}$ is already in simplest form.

5. Divide:

   ${ w = \frac{(3x - 2)(2x + 5)}{2x + 5} }$

6. Cancel common factors: Cancel ${(2x + 5)}$:

   ${ w = 3x - 2 }$

So, the width is ${3x - 2}$.

Problem 2

The area of a rectangle is ${A = 15x^2 - x - 2}$, and the length is ${l = 3x + 1}$. Find the width ${w}$.

Solution

1. Write the formula: ${w = \frac{A}{l}}$

2. Substitute:

   ${ w = \frac{15x^2 - x - 2}{3x + 1} }$

3. Factor the area: We need to factor ${15x^2 - x - 2}$. Look for two numbers that multiply to ${15 \times -2 = -30}$ and add up to -1. Those numbers are -6 and 5. So, we rewrite the middle term:

   ${ 15x^2 - x - 2 = 15x^2 - 6x + 5x - 2 }$

   Factor by grouping:

   ${ (15x^2 - 6x) + (5x - 2) = 3x(5x - 2) + 1(5x - 2) }$

   Factor out the common term ${(5x - 2)}$:

   ${ 3x(5x - 2) + 1(5x - 2) = (3x + 1)(5x - 2) }$

4. Factor the length: The length ${3x + 1}$ is already in simplest form.

5. Divide:

   ${ w = \frac{(3x + 1)(5x - 2)}{3x + 1} }$

6. Cancel common factors: Cancel ${(3x + 1)}$:

   ${ w = 5x - 2 }$

So, the width is ${5x - 2}$.

These practice problems should give you a good handle on finding the width of a rectangle when you know its area and length. Remember, the key is to factor the expressions carefully and cancel common factors. Keep up the great work, guys!

Conclusion

Alright, we’ve reached the end of our journey on finding the width of a rectangle given its area and length. You've learned how to tackle these problems step by step, from setting up the formula to factoring and simplifying. You've also seen how this skill applies in real-world scenarios, making it more than just a math problem.

Remember, the key takeaways are:

• Use the formula ${w = \frac{A}{l}}$ to find the width.
• Factor both the area and the length expressions whenever possible.
• Cancel common factors to simplify the expression.
• Simplify your final answer.
• Avoid common mistakes like forgetting to factor, incorrect factoring, and sign errors.

By mastering these steps, you’ll be well-prepared to solve similar problems with confidence. Keep practicing, and you'll find that these calculations become second nature. Whether you're working on homework, planning a home renovation, or just want to flex your math muscles, you’ve now got the tools you need.

So, go ahead and apply what you’ve learned. Try solving more problems, explore different scenarios, and see how this knowledge fits into other areas of math and real-life situations. You’ve got this, guys! Keep up the awesome work, and happy calculating!