Solving For The Width Of A Rectangular Pan A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like it's written in another language? Don't worry, we've all been there. Let's break down this seemingly complex question about a rectangular pan and figure out the width. We'll take it slow and make sure everyone's on the same page. So, grab your thinking caps, and let's dive in!

The Problem: Decoding the Dimensions

Let's first understand the problem, rectangular pan dimensions. We're told we have a rectangular pan where the length is $\frac{4}{3}$ times its width. Think of it like this: if the width was 3 inches, the length would be 4 inches. The area, which is the space inside the pan, is 432 square inches. Our mission, should we choose to accept it, is to find the width of this pan.

Before we jump into calculations, let's recap the key information. The length is related to the width by a fraction, and we know the total area. The big question is how to use this information to unlock the width. We'll need to dust off some geometry basics, but don't sweat it – we'll walk through it together. Remember, math problems are just puzzles waiting to be solved! The key to unraveling this puzzle lies in translating the words into mathematical expressions. Once we have those expressions, the rest will fall into place more easily.

We'll be using the formula for the area of a rectangle, which is something most of us probably learned way back in elementary school. It's a simple formula, but it's incredibly powerful for solving problems like this one. So, keep that formula in mind as we move forward. We are going to rewrite the relationships given in the problem using math, and then use basic algebra to solve it. Stay with me, and we'll conquer this problem together.

Setting Up the Equation: Math to the Rescue

The cornerstone of our solution lies in translating the word problem into a mathematical equation, and to effectively accomplish this, we'll introduce a variable. Let's designate the width of the pan as "w". This simple step allows us to represent the unknown quantity in a tangible way. Now, recall that the problem states the length is $\frac{4}{3}$ times the width. Using our variable, we can express the length as $\frac{4}{3}w$. This is where the problem starts to take shape algebraically.

Remember the area of a rectangle is length times width? This is critical information. We know the area is 432 square inches, and we have expressions for both the length and width in terms of w. Now, we can formulate an equation: Area = Length × Width becomes $432 = (\frac{4}{3}w) * w$. See how we've taken the given information and transformed it into a concise mathematical statement? This is a crucial step in problem-solving. By converting the word problem into an equation, we've set the stage for using algebraic techniques to find the solution.

Now, we have a clear equation that represents the relationships described in the problem. The next step involves simplifying this equation to isolate our variable, w. This is where our algebra skills will come into play. We'll need to combine like terms, and perform some strategic manipulations to get w by itself. So, let's prepare to roll up our sleeves and dive into the algebraic manipulation that will reveal the width of the pan.

Solving for the Width: Algebra in Action

Now, let's simplify the equation $432 = (\frac{4}{3}w) * w$. Multiplying the terms on the right side gives us $432 = \frac{4}{3}w^2$. Our goal is to isolate $w^2$, so we need to get rid of the fraction. We can do this by multiplying both sides of the equation by $\frac{3}{4}$. This is a crucial algebraic step that maintains the balance of the equation while moving us closer to our solution.

Multiplying both sides by $\frac{3}{4}$ gives us $(\frac{3}{4}) * 432 = w^2$. If we do the math, $(\frac{3}{4}) * 432$ equals 324. So, our equation now looks like this: $324 = w^2$. We're getting closer! We have $w^2$ isolated, but we want w. To undo the squaring, we need to take the square root of both sides. Remember that the square root of a number is a value that, when multiplied by itself, equals the original number. This is a fundamental concept in algebra. The square root operation is the inverse of the squaring operation, allowing us to peel away the exponent and reveal the value of w.

Taking the square root of both sides, we get $w = \sqrt{324}$. Now, we need to figure out what the square root of 324 is. If you know your squares, you might recognize this one. If not, don't worry, you can use a calculator or try to find a number that, when multiplied by itself, gives you 324. The square root of 324 is 18. So, $w = 18$. We've found it! The width of the pan is 18 inches.

The Answer and Its Significance

So, the width of the cake pan is 18 inches. That's option C! Awesome job, guys! You've successfully navigated through the problem, translated words into math, and used algebra to find the answer. But hold on, we're not just about getting the right answer; we want to understand what it means.

Think about it: we started with a description of a rectangular pan, a relationship between its length and width, and its total area. We used this information to calculate the width. This shows how math can be used to solve real-world problems, even something as simple as figuring out the dimensions of a baking pan. The application of mathematical principles extends far beyond the classroom, permeating various aspects of our daily lives. From calculating measurements in cooking to optimizing dimensions in construction, mathematical skills are invaluable tools for problem-solving in practical contexts.

Furthermore, this problem highlights the power of algebra. By using variables and equations, we can represent unknown quantities and solve for them. This is a fundamental skill in mathematics and is used in countless applications in science, engineering, and many other fields. So, this wasn't just about finding the width of a pan; it was about using math to make sense of the world around us. We were able to convert the word problem into an algebraic equation, manipulate the equation using algebraic principles, and find the value of our unknown variable.

Key Takeaways and Practice

Let's recap the key steps we took to solve this problem, we have translated word problems into mathematical equations. This is a crucial skill in mathematics. We have learned how to use variables to represent unknown quantities and form equations based on the given information. Remember, the more you practice, the better you'll become at this. When facing a word problem, first read it carefully and identify the key information. What are you trying to find? What relationships are given?

Next, translate the words into mathematical expressions and equations. This often involves assigning variables to unknown quantities. Once you have an equation, use your algebra skills to solve for the unknown variable. Remember to check your answer to make sure it makes sense in the context of the problem. We've also applied the formula for the area of a rectangle. Knowing basic geometric formulas is essential for solving many math problems. If you're not familiar with these formulas, take some time to review them.

Finally, we have used algebraic techniques to solve the equation. This involved simplifying the equation, isolating the variable, and taking the square root. Practice these techniques, and you'll become more confident in your algebra skills. To solidify your understanding, try solving similar problems. You can find practice problems in textbooks, online resources, or even create your own problems. The more you practice, the more natural these steps will become. And remember, don't be afraid to ask for help if you get stuck. Math can be challenging, but it's also incredibly rewarding. Keep practicing, and you'll become a math master in no time!

Wrapping Up: You've Got This!

So, there you have it! We've successfully solved for the width of the rectangular pan. Remember, tackling math problems is like building a muscle – the more you work at it, the stronger you become. Don't be discouraged by challenging problems; break them down into smaller steps, and you'll be surprised at what you can achieve. You guys are awesome, and with a little practice, you can conquer any math problem that comes your way! Keep up the great work!