Rectangle Width Calculation: Perimeter And Length

Hey everyone! Let's dive into a cool math problem today. We're going to figure out how to find the width of a rectangle when we know its perimeter and length. It might sound tricky at first, but trust me, it's totally manageable. We will break it down into easy-to-understand steps, no sweat! This is a classic example of using algebraic expressions and a bit of problem-solving. So, let's get started and make some math magic!

Understanding the Basics: Perimeter and Rectangles

Alright, before we jump into the problem, let's make sure we're all on the same page. Remember what a rectangle is, right? It's a four-sided shape where all the angles are right angles (90 degrees), and the opposite sides are equal in length. Now, the perimeter of a rectangle is just the total distance around its outside. Imagine you're walking around the rectangle – the perimeter is the total distance you'd walk. It's like putting a fence around your rectangular garden; the perimeter is the length of the fence you'd need.

The formula for the perimeter of a rectangle is pretty straightforward: Perimeter = 2 * (length + width). This means we add the length and width together, and then multiply the sum by 2. It takes into account that there are two lengths and two widths in every rectangle. Got it? Awesome! The length and width are crucial because they define the size and shape of the rectangle. The length is usually considered the longer side, and the width is the shorter side, although, technically, you can call them whatever you want!

So, in this problem, we're given the perimeter as an algebraic expression and the length as another algebraic expression. Our goal is to find an expression for the width. This involves using the perimeter formula and some algebra to solve for the width. We're going to utilize the relationships between perimeter, length, and width to determine the missing side. This process involves substitution and simplification – fundamental tools in algebra. Remember that algebraic expressions contain variables and constants, and we'll be manipulating these to find our answer. Don't worry, it's less complicated than it sounds!

The Given Information and Our Goal

Okay, let's get down to the nitty-gritty of our problem. We're told that the perimeter of a rectangle is given by the expression: $26c^2 + 26c - 18$. This is a quadratic expression, meaning it involves a variable (c) raised to the power of 2. We're also given the length of the rectangle: $11c^2 - 10$. Our mission, should we choose to accept it, is to find an expression that represents the width of the rectangle. So, we're not looking for a numerical answer, but another algebraic expression. It's like building a puzzle. We have the perimeter, which tells us the total “frame” of the rectangle, and one side (the length). Our task is to find the other side (the width).

Think of it this way: We have the “total” distance around the rectangle and the distance of one of the sides. We need to work backward to find the length of the adjacent side. This type of problem requires you to use the perimeter formula and your knowledge of algebraic manipulations. We will use the formula: Perimeter = 2 * (length + width), and rearrange it to solve for the width. Understanding how to rearrange an equation to solve for a specific variable is a super useful skill in algebra.

We know the perimeter and the length; therefore, we can plug these values into the formula and solve for the width. The process will involve substituting the given expressions into the perimeter formula and simplifying the resulting equation. Keep in mind that when we're dealing with algebraic expressions, the goal is to combine like terms and isolate the variable we're solving for. So, are you ready to dive in and find that width? Let's go!

Solving for the Width: Step-by-Step

Alright, time to roll up our sleeves and get to work! We know the perimeter formula: Perimeter = 2 * (length + width). We also know that the perimeter is $26c^2 + 26c - 18$ and the length is $11c^2 - 10$. Let's call the width "w" for simplicity. First, substitute the known values into the formula: $26c^2 + 26c - 18 = 2 * ((11c^2 - 10) + w)$. Our next step is to simplify the equation. Divide both sides by 2 to get rid of the 2 on the right side: $(26c^2 + 26c - 18) / 2 = (11c^2 - 10) + w$.

When we divide, we need to divide each term in the perimeter expression by 2: $13c^2 + 13c - 9 = 11c^2 - 10 + w$. Now, our goal is to isolate "w" on one side of the equation. To do this, subtract $(11c^2 - 10)$ from both sides: $13c^2 + 13c - 9 - (11c^2 - 10) = w$. Simplify by subtracting like terms: $(13c^2 - 11c^2) + 13c + (-9 + 10) = w$. This gives us: $2c^2 + 13c + 1 = w$.

Therefore, the expression for the width of the rectangle is $2c^2 + 13c + 1$. Awesome! We've successfully found the width. This result gives us a general expression for the width, which is defined in terms of "c." You can use this expression to calculate the width of the rectangle if you have the value of "c." Remember, this value of “c” could be any real number. So, for different values of "c," you will get different dimensions. Congrats on sticking with it!

Checking Our Work and Interpretation

It's always a good idea to double-check your work, right? So, let's see if our answer makes sense. We found that the width is $2c^2 + 13c + 1$, the length is $11c^2 - 10$, and we can calculate the perimeter based on the answer. According to the original formula: Perimeter = 2 * (length + width). So: Perimeter = 2 * (($11c^2 - 10$) + ($2c^2 + 13c + 1$)). Simplify the terms inside the parentheses: Perimeter = 2 * ($13c^2 + 13c - 9$). Distribute the 2: Perimeter = $26c^2 + 26c - 18$. This matches the perimeter that we were given in the beginning of the problem! Awesome! This confirms our answer for the width is correct.

So, what does this expression for the width mean in the real world? Well, it tells us how the width of the rectangle changes as the value of "c" changes. The "c" is like a variable scale factor. When "c" is zero, the width is 1. When "c" is a positive number, the width increases or decreases depending on the value. Keep in mind that the width is an algebraic expression, not a single numerical value. It's a formula, and, as we said, we can plug in a value for "c" to get a specific width. So, the width of the rectangle depends on the value of "c." The relationships between these variables show a beautiful aspect of algebra – you can describe shapes and their properties using variables and equations. Isn't math cool?

Conclusion: Wrapping Things Up

That's a wrap, guys! We've successfully found the expression for the width of a rectangle given its perimeter and length. We started with the perimeter formula, plugged in our known values, and used some basic algebra to solve for the width. We also took the time to check our work to ensure everything made sense. You did a great job following along!

This problem showed us the power of using algebraic expressions and formulas to solve real-world problems. Whether you're dealing with perimeters, areas, or any other geometric shapes, understanding the formulas and how to manipulate them is super important. Remember, math is like a puzzle; sometimes, you have to rearrange the pieces to get the full picture. Keep practicing, keep learning, and don't be afraid to tackle new challenges.

So, the next time you encounter a problem like this, you'll know exactly what to do. You can find the unknown dimensions of a shape with the help of the known measures and formulas. Keep in mind the relationship between shapes and equations. The more you practice, the more confident you'll become. And always remember to double-check your work! Until next time, keep exploring the amazing world of mathematics!