Unveiling Rectangle Dimensions: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into a classic geometry problem: figuring out the dimensions of a rectangle. We're given some clues – the relationship between the length and width, and the area – and our mission is to crack the code and find the actual measurements. So, grab your pencils, get your thinking caps on, and let's unravel this geometric puzzle together! This problem is a fantastic example of how algebra and geometry work hand-in-hand to solve real-world problems. We'll be using the power of equations to represent the information we have and then solving those equations to find the unknown values. It's like being a detective, but instead of finding clues, we're finding the dimensions of a shape. We'll break down the problem step-by-step, making sure every concept is clear and easy to follow. Along the way, we will cover all the essential aspects and elements necessary to arrive at the solution. This will help you become more comfortable with this type of problem, improving your math skills and boosting your confidence. So, let's begin and get ready to be amazed at how easily we can solve this problem! This is a great opportunity to explore the relationship between the area, length, and width of rectangles. We'll see how changing one dimension affects the other, and how the area remains constant. By the end, you'll be able to tackle similar problems with ease and have a solid understanding of this geometric concept. Understanding how to solve this will give you a solid foundation for more complex mathematical concepts. This is like building the foundation of a house; without it, the whole structure will be unstable. That means this is a good first step to your success in mathematics.
Setting Up the Problem: Understanding the Clues
First, let's break down the information we have. We're told that "the length of a rectangle is 5 centimeters less than three times its width". This statement gives us a direct relationship between the length and the width. We can translate this into an algebraic expression. Let's use w to represent the width and l to represent the length. We also know that "its area is 12 square centimeters". The area of a rectangle is calculated by multiplying its length by its width (Area = length * width). This gives us a second piece of information. Remember, the area is a measure of the space inside the rectangle, so it's measured in square units. So we have all the data to successfully solve this question. We'll be using this information to create equations that we can solve. The key to solving these kinds of problems is to accurately translate the word problem into a mathematical equation. Careful reading and understanding each piece of information will prevent any mistakes. A common mistake is to confuse the length and width in the equations or not properly express the given relationships. So, take your time, and write down each element of information carefully. Once you have everything written down and have a solid understanding of each part of the problem, the solution becomes much more simple. By carefully translating each sentence of the question, we can create the proper equations to begin with. This problem is not difficult; it only requires us to properly structure the question.
Translating Words into Equations: The Algebraic Approach
Let's turn those words into mathematical equations. We know:
- Length: l = 3w - 5 (The length is three times the width minus 5).
- Area: l * w* = 12 (The length multiplied by the width equals 12).
Now, we have two equations, and we can use these equations to solve for the width and the length. Since we know what l is equal to (3w - 5), we can substitute that value into the area equation. This method is called substitution, and it's a powerful tool in algebra. After substituting the equation for length, our area equation becomes: (3w - 5) * w* = 12. Notice how this new equation only has w in it. We can now solve for w. This will give us the width of the rectangle. After we find the width, we can use the first equation to find the length. This will allow us to find our solution! See, this isn't that difficult, right? The most important part of solving these questions is proper equation translation. Now we can simplify and solve the quadratic equation to find the values of w. Remember that practice makes perfect, and with each problem you solve, you will get better at setting up and solving these kinds of equations.
Solving for Width: Finding w
Let's solve the equation (3w - 5) * w* = 12. First, we need to distribute the w: 3w² - 5w = 12. Next, we need to set the equation to zero to solve it. Subtract 12 from both sides to get: 3w² - 5w - 12 = 0. This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring works well. We're looking for two numbers that multiply to give us -36 (3 * -12) and add up to -5. Those numbers are -9 and 4. So, we rewrite the middle term and factor by grouping: 3w² - 9w + 4w - 12 = 0. Factor out 3w from the first two terms and 4 from the last two: 3w(w - 3) + 4(w - 3) = 0. Now factor out (w - 3): (w - 3)(3w + 4) = 0. This gives us two possible solutions for w: w = 3 or w = -4/3. Since the width of a rectangle cannot be negative, we discard the negative solution. Therefore, w = 3 centimeters. So, the width of the rectangle is 3 centimeters. We've used algebra, factored an equation, and found a reasonable answer! We have now found our first answer, so let's continue. We are almost at the end of this problem, and it is going great. We have taken the right steps and have done everything properly. Now that we have the width, we can move forward and find the length of the rectangle.
Finding the Length: Calculating l
Now that we know the width (w) is 3 centimeters, we can find the length (l) using the equation l = 3w - 5. Substitute w = 3 into the equation: l = 3 * 3 - 5. l = 9 - 5. l = 4 centimeters. The length of the rectangle is 4 centimeters. We can now see that our answer makes perfect sense! The length is 5 centimeters less than 3 times the width (3 * 3 = 9, and 9 - 5 = 4). We can now say we have the final answers for our question. Now that we have both the width and the length, we've successfully found the dimensions of the rectangle. Let's make sure our answer is correct by verifying the area. Remember, the area is length times width. So, 4 cm * 3 cm = 12 square centimeters. This matches the information given in the problem, so we can be confident in our solution. This process of verifying your answer is an essential step in math. It allows you to check for any potential errors and ensure you have understood the problem and have solved it correctly.
The Answer: Dimensions of the Rectangle
So, the dimensions of the rectangle are:
- Width: 3 centimeters
- Length: 4 centimeters
Congratulations! We've successfully solved the problem. It's awesome, right? We started with the problem, translated it into equations, solved those equations, and arrived at the dimensions of the rectangle. This process is a testament to the power of mathematics. By combining algebraic knowledge with geometric principles, we can solve problems that might seem complex at first glance. Remember that each step is designed to help you understand the problem better. Now, the more practice you do, the easier it will become. You will be able to do this type of problem very quickly. This entire process demonstrates that problem-solving in mathematics is not just about memorizing formulas; it is about understanding concepts and applying them creatively. The more you explore, the more confident you become in your abilities.
Conclusion: Mastering the Rectangle's Dimensions
We did it! We successfully found the dimensions of the rectangle! This process shows how you can use algebra to solve geometric problems. By breaking down the problem into smaller parts, translating words into equations, and applying your algebraic skills, you can find the solution. Remember, practice is essential. The more problems you solve, the more comfortable you will become with these types of questions. This kind of problem-solving helps improve your math skills, boosts your confidence, and prepares you for more complex mathematical concepts in the future. Now go out there and apply your new skills. This ability will help you in your school and future life! Keep up the great work, and continue to explore the fascinating world of mathematics. Keep practicing and exploring the exciting realm of math! You're well on your way to becoming a math whiz. With each problem you solve, you are building a stronger understanding and a more powerful skill set. Mathematics is a journey, not a destination, so keep learning and exploring!
For further understanding of related topics, you can check out resources from Khan Academy (https://www.khanacademy.org/) for detailed explanations and additional practice problems. You can also explore Math is Fun (https://www.mathsisfun.com/) to enrich your understanding of different mathematical concepts and improve your skills.
