Solving For Rectangle Length With Given Area And Width Relationship

by Omar Yusuf 68 views

Hey guys! Today, let's dive into a classic math problem that involves finding the dimensions of a rectangle when we know its area and a relationship between its length and width. This type of problem often appears in algebra and geometry, and it's a great way to sharpen our problem-solving skills. We'll break down the problem step-by-step, using clear explanations and examples, so you can confidently tackle similar questions in the future. So, grab your pencils and let's get started!

Problem Statement

The width of a rectangle is 3 less than the length. The area of the rectangle is 88. What is the length of the rectangle?

This is a classic word problem that requires us to translate the given information into algebraic equations. We're told about the relationship between the width and length, and we're given the area. Our mission is to find the length. To do this effectively, we'll use the formula for the area of a rectangle and some algebraic manipulation. Let's break it down.

Setting Up the Equations

The key to solving this problem is to translate the words into mathematical expressions. This involves identifying the variables, establishing their relationships, and forming equations. So, let's start by defining our variables:

  • Let's use 'L' to represent the length of the rectangle.
  • Since the width is 3 less than the length, we can represent the width as 'L - 3'.

Now, we know the area of a rectangle is calculated by multiplying its length and width. So, we can write the equation for the area as:

Area = Length × Width

We're given that the area is 88, so we can substitute that into the equation:

88 = L × (L - 3)

This is a quadratic equation, and solving it will give us the length of the rectangle. Quadratic equations might seem intimidating, but don't worry! We'll go through the steps together.

Solving the Quadratic Equation

Okay, so we have our equation: 88 = L × (L - 3). Now, we need to solve for L. The first step is to expand the equation by multiplying L with the terms inside the parentheses:

88 = L² - 3L

To solve a quadratic equation, we need to set it equal to zero. So, let's subtract 88 from both sides:

0 = L² - 3L - 88

Now, we have a standard quadratic equation in the form of ax² + bx + c = 0. There are a few ways to solve this, but factoring is often the quickest method if it's possible. Factoring involves finding two numbers that multiply to give 'c' (-88 in our case) and add up to give 'b' (-3 in our case). Those numbers are -11 and 8, because -11 × 8 = -88 and -11 + 8 = -3. So, we can rewrite the equation as:

0 = (L - 11)(L + 8)

Now, for the product of two factors to be zero, at least one of them must be zero. So, we set each factor equal to zero:

L - 11 = 0 or L + 8 = 0

Solving these equations gives us two possible values for L:

L = 11 or L = -8

Interpreting the Solution

We've found two possible values for the length: 11 and -8. But wait a minute! Can the length of a rectangle be negative? Nope! In the real world, dimensions like length and width can't be negative. So, we discard the solution L = -8. That leaves us with:

L = 11

So, the length of the rectangle is 11 units. Now, let's find the width. Remember, the width is 3 less than the length, so:

Width = L - 3 = 11 - 3 = 8

So, the width of the rectangle is 8 units.

Verifying the Solution

It's always a good idea to check our answer to make sure it's correct. We can do this by plugging the values we found for length and width back into the original problem. We found that the length is 11 and the width is 8. Let's check if the area is indeed 88:

Area = Length × Width = 11 × 8 = 88

Great! Our answer checks out. The area is 88, as given in the problem. This gives us confidence that we've solved the problem correctly.

Alternative Methods

While we solved this problem by factoring the quadratic equation, there are other methods we could have used. One common method is the quadratic formula. The quadratic formula is a general formula that can be used to solve any quadratic equation in the form ax² + bx + c = 0. The formula is:

L = (-b ± √(b² - 4ac)) / (2a)

In our equation, L² - 3L - 88 = 0, a = 1, b = -3, and c = -88. Plugging these values into the quadratic formula would also give us the solutions L = 11 and L = -8. We would still discard the negative solution for the same reason as before.

Another method is completing the square. Completing the square involves manipulating the quadratic equation to create a perfect square trinomial. This method can be a bit more involved, but it's a valuable technique to know for solving quadratic equations.

Common Mistakes to Avoid

When solving problems like this, it's easy to make small mistakes that can lead to incorrect answers. Here are some common pitfalls to watch out for:

  1. Incorrectly Setting Up the Equation: The most common mistake is misinterpreting the relationship between the length and width. Make sure you correctly translate the words into algebraic expressions. For example, if the problem says the width is 3 less than the length, it's important to represent the width as L - 3, not 3 - L.
  2. Sign Errors: When expanding and rearranging the equation, be careful with signs. A simple sign error can throw off your entire solution.
  3. Forgetting to Set the Equation to Zero: Remember, to solve a quadratic equation by factoring, you need to set it equal to zero first.
  4. Incorrectly Factoring: Factoring can be tricky, especially with larger numbers. Double-check your factors to make sure they multiply to give the correct constant term and add up to give the correct coefficient of the linear term.
  5. Not Discarding the Negative Solution: In real-world problems involving dimensions, negative solutions don't make sense. Always discard negative solutions if they don't fit the context of the problem.
  6. Not Verifying the Solution: Always take the time to check your answer by plugging it back into the original problem. This can help you catch mistakes and ensure that your solution is correct.

Real-World Applications

Solving problems involving the dimensions of rectangles might seem like just a math exercise, but it has many real-world applications. These skills are useful in various fields, including:

  1. Construction and Architecture: When designing buildings or structures, architects and engineers often need to calculate areas and dimensions to ensure proper space planning and material usage.
  2. Interior Design: Interior designers use these skills to arrange furniture, plan room layouts, and determine the amount of flooring or wallpaper needed.
  3. Gardening and Landscaping: Gardeners and landscapers use area calculations to plan garden layouts, estimate the amount of soil or mulch needed, and determine the size of flower beds or lawns.
  4. Real Estate: Real estate agents and buyers often calculate areas of properties to determine their value and compare different options.
  5. Manufacturing and Engineering: Engineers and manufacturers use these calculations to design and produce various products, from electronic devices to automobiles.

Practice Problems

To really master these concepts, it's important to practice! Here are a few more problems similar to the one we just solved. Try working through them on your own, and don't be afraid to review the steps we discussed if you get stuck.

  1. The width of a rectangle is 5 less than the length. The area of the rectangle is 104. What is the length of the rectangle?
  2. The length of a rectangle is 7 more than the width. The area of the rectangle is 144. What is the width of the rectangle?
  3. The width of a rectangle is half the length. The area of the rectangle is 72. What are the dimensions of the rectangle?

Conclusion

So, guys, we've successfully tackled a problem involving finding the length of a rectangle when we know its area and the relationship between its length and width. We broke down the problem into manageable steps, from setting up the equations to solving the quadratic equation and interpreting the solution. We also discussed common mistakes to avoid and real-world applications of these skills. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! Happy problem-solving!