Solving For Rectangle Length: An In-Depth Mathematical Guide
Understanding how to determine the dimensions of geometric shapes is a fundamental concept in mathematics. In this article, we will delve into a specific problem involving a rectangle, where the width is defined in terms of its length, and the area is known. Our main goal is to identify the correct equation that can be used to calculate the length of this rectangle. This exploration will not only reinforce your understanding of basic geometry but also enhance your problem-solving skills. Let's break down the problem step by step and arrive at the solution.
Problem Statement
The problem presents a scenario where a rectangle's width is 2 meters shorter than its length, which we'll denote as I. The area of the rectangle is given as 168 square meters. The central question is: Which equation can be used to find the length of the rectangle? This type of problem is a classic application of algebraic principles in geometry, requiring a clear understanding of the relationships between the dimensions and area of a rectangle.
Key Concepts: Area of a Rectangle
Before diving into the solution, let's revisit the basic formula for the area of a rectangle. The area of a rectangle is calculated by multiplying its length by its width. Mathematically, this is expressed as:
Area = Length × Width
This formula is the cornerstone of our approach to solving this problem. We need to express both the length and the width in terms of a single variable and then use the given area to form an equation.
Defining the Variables
In this problem, the length of the rectangle is given as I. The width is described as 2 meters shorter than the length. Therefore, we can express the width in terms of I as:
Width = I - 2
Now that we have both the length and the width expressed in terms of I, we can use the area formula to set up an equation.
Setting Up the Equation
We know that the area of the rectangle is 168 square meters. Using the formula for the area of a rectangle, we can write:
Area = Length × Width
Substituting the expressions for length and width, we get:
168 = I × (I - 2)
This equation represents the relationship between the length I and the area of the rectangle. Expanding this equation will give us a quadratic equation that we can solve to find the value of I.
Expanding the Equation
To better understand the equation and to match it with the given options, let's expand the equation:
168 = I × (I - 2)
168 = I² - 2I
This expanded form is a quadratic equation, which can be rearranged into the standard form:
I² - 2I - 168 = 0
Now, let's compare this equation with the options provided in the problem statement.
Analyzing the Options
The problem presents two possible equations:
- I(I - 2) = 168
- I(I + 2) = 168
Our derived equation, 168 = I × (I - 2), directly matches the first option. This equation accurately represents the given conditions: the length I multiplied by the width (I - 2) equals the area 168 square meters.
Why the Other Option is Incorrect
The second option, I(I + 2) = 168, implies that the width is 2 meters longer than the length, which contradicts the problem statement. The problem clearly states that the width is 2 meters shorter than the length. Therefore, this equation does not accurately represent the given scenario.
Conclusion: The Correct Equation
After carefully analyzing the problem and deriving the equation based on the given information, we can confidently conclude that the correct equation to find the length of the rectangle is:
I(I - 2) = 168
This equation accurately captures the relationship between the length, width, and area of the rectangle, as described in the problem statement. Solving this equation will yield the possible values for the length I, from which we can choose the appropriate solution (a positive value).
Solving the Equation and Finding the Length
To further illustrate the process, let's solve the equation I(I - 2) = 168 to find the length I. We've already expanded the equation to its quadratic form:
I² - 2I - 168 = 0
This is a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = -2, and c = -168. We can solve this equation using various methods, such as factoring, completing the square, or using the quadratic formula.
Solving by Factoring
Factoring involves finding two numbers that multiply to -168 and add up to -2. These numbers are -14 and 12. So, we can rewrite the quadratic equation as:
(I - 14)(I + 12) = 0
Setting each factor equal to zero gives us two possible solutions for I:
I - 14 = 0 or I + 12 = 0
I = 14 or I = -12
Choosing the Correct Solution
Since length cannot be negative, we discard the solution I = -12. Therefore, the length of the rectangle is I = 14 meters.
Determining the Width
Now that we have the length, we can find the width using the expression we derived earlier:
Width = I - 2
Substituting I = 14, we get:
Width = 14 - 2 = 12 meters
So, the dimensions of the rectangle are 14 meters in length and 12 meters in width.
Verification
To verify our solution, we can multiply the length and width to ensure we get the given area:
Area = Length × Width
Area = 14 meters × 12 meters = 168 square meters
This confirms that our solution is correct. The length of the rectangle is 14 meters, and the width is 12 meters.
Importance of Accurate Equations in Problem Solving
This problem highlights the critical role of setting up the correct equation. An accurate equation is the foundation for solving any mathematical problem. In this case, understanding the relationship between the length, width, and area of a rectangle was crucial in forming the correct equation. A slight misinterpretation of the problem statement or an incorrect application of the formula could lead to a wrong equation and, consequently, an incorrect solution.
Common Mistakes and How to Avoid Them
One common mistake in such problems is misinterpreting the relationship between the dimensions. For example, confusing “2 meters shorter” with “2 meters longer” would lead to using the wrong expression for the width (I + 2 instead of I - 2). To avoid this, it’s essential to read the problem statement carefully and underline or highlight key phrases.
Another common mistake is an error in algebraic manipulation, such as incorrectly expanding or factoring the quadratic equation. Regular practice and a thorough understanding of algebraic principles can help prevent these errors.
The Significance of Verification
Verifying the solution is a crucial step in the problem-solving process. It helps to ensure that the answer is correct and that no mistakes were made along the way. In this case, we verified our solution by multiplying the length and width to confirm that the result matched the given area. This step provides confidence in the accuracy of the solution.
Real-World Applications
The concepts and skills used in this problem are applicable in various real-world scenarios. Understanding how to calculate areas and dimensions is essential in fields such as architecture, engineering, construction, and interior design. For example, architects need to calculate the dimensions of rooms and buildings, engineers need to determine the size of structures, and interior designers need to plan the layout of spaces.
Practical Examples
Consider a scenario where you are designing a rectangular garden. You know the total area you want the garden to cover and have a constraint on the relationship between the length and width (e.g., the width must be a certain amount shorter than the length). The same principles we used in this problem can be applied to determine the dimensions of the garden.
Another example is in manufacturing, where companies need to calculate the dimensions of materials to minimize waste and optimize production. Understanding how to set up and solve equations involving areas and dimensions is a valuable skill in these contexts.
Expanding the Problem
To further challenge your understanding, let's consider a variation of the problem. Suppose the perimeter of the rectangle is also given, in addition to the area. How would you approach this problem? Let’s explore this extended scenario.
Introducing the Perimeter
The perimeter of a rectangle is the total distance around its sides, which is calculated as:
Perimeter = 2 × (Length + Width)
Let's say the perimeter of the rectangle in our problem is 52 meters. Now we have two pieces of information:
- Area = 168 square meters
- Perimeter = 52 meters
We already know that the width is I - 2. Using the perimeter formula, we can write:
52 = 2 × (I + I - 2)
Solving the System of Equations
Now we have a system of two equations:
- I(I - 2) = 168
- 52 = 2 × (2I - 2)
The second equation can be simplified to:
26 = 2I - 2
2I = 28
I = 14
This confirms our previous result for the length. We can then find the width as before:
Width = I - 2 = 14 - 2 = 12 meters
This extended problem demonstrates how additional information can be incorporated to reinforce the solution and provide further verification.
Conclusion: Mastering Geometric Problems
In conclusion, finding the correct equation to solve a geometric problem requires a clear understanding of the underlying principles and relationships. In the case of a rectangle, the relationship between the length, width, and area is fundamental. By carefully defining the variables, setting up the equation, and verifying the solution, we can confidently solve such problems. The ability to solve these types of problems is not only valuable in mathematics but also in various real-world applications. Whether you are designing a garden, planning a building, or optimizing a manufacturing process, the skills you develop in geometry and algebra are essential for success. Remember to always read the problem statement carefully, define your variables clearly, and verify your solution to ensure accuracy.
By working through this problem and its extensions, you have strengthened your problem-solving skills and gained a deeper understanding of geometric principles. Keep practicing, and you’ll be well-equipped to tackle even more challenging problems in the future.
This article provides a detailed explanation of how to determine the correct equation for finding the length of a rectangle, given its area and the relationship between its length and width. We break down the problem step by step, highlighting the importance of understanding geometric principles and algebraic techniques. Our goal is to help readers master these essential skills for problem-solving in mathematics and real-world applications.
Understanding the Problem: Setting Up the Foundation
At the heart of our discussion is the problem of finding the equation to determine the length of a rectangle. The key to solving this problem lies in a clear understanding of the given information and how it relates to the properties of a rectangle. Rectangle problems often involve understanding the relationship between the length, width, and area, and setting up the correct equation is the first crucial step. The given problem states that the width of a rectangle is 2 meters shorter than its length (I), and its area is 168 square meters. To find the equation that can be used to determine the length of the rectangle, we need to express the dimensions and area in mathematical terms. This means we need to define the variables and establish the relationships between them. Let's first define the variables. We denote the length of the rectangle as I. Since the width is 2 meters shorter than the length, we express the width as I - 2. The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is given as 168 square meters. Therefore, we can set up the equation: 168 = I × (I - 2). This equation is the foundation for solving the problem. It mathematically represents the relationship between the length, width, and area of the rectangle. The next step is to analyze and simplify this equation to match one of the options provided. To ensure the accuracy of our setup, let’s briefly discuss common mistakes that students make when encountering such problems. One frequent error is misinterpreting the relationship between the length and the width. For instance, students may mistakenly add 2 meters instead of subtracting it from the length to find the width. This misunderstanding can lead to an incorrect equation and, subsequently, an incorrect solution. Another common error is not correctly applying the formula for the area of a rectangle. Some students might confuse it with the formula for the perimeter, which is 2(Length + Width). Therefore, it is crucial to ensure that the correct formula is used to set up the equation. By paying attention to these details and setting up the equation correctly, we lay the groundwork for an accurate solution. The following sections will delve into the steps of simplifying the equation and finding the length of the rectangle.
Forming the Equation: Connecting Geometry and Algebra
In this section, we focus on forming the algebraic equation that connects the geometric properties of the rectangle with the given information. This is a crucial step because the correct equation is the foundation for solving the problem. As we established earlier, the area of a rectangle is calculated by multiplying its length by its width. We know that the length is I meters, the width is (I - 2) meters, and the area is 168 square meters. The formula for the area of a rectangle is: Area = Length × Width. Substituting the known values into this formula, we get: 168 = I × (I - 2). This equation is the heart of our problem. It mathematically describes the relationship between the length (I), the width (I - 2), and the area (168) of the rectangle. Our task now is to analyze this equation and determine which of the given options matches this representation. To gain a deeper understanding of this equation, let's consider what it represents in practical terms. The term I represents the length of the rectangle, and (I - 2) represents the width. When we multiply these two terms together, we are calculating the area enclosed within the rectangle. The result of this multiplication must equal 168 square meters, which is the given area. This equation is a quadratic equation in disguise. To see this more clearly, we can expand the equation: 168 = I × (I - 2) becomes 168 = I² - 2I. This form of the equation reveals its quadratic nature and shows that there will be two possible solutions for I. However, in the context of this problem, only the positive solution makes sense, as length cannot be negative. Now, let's consider how to analyze the given options and determine which one is correct. The options will likely be in a similar form to the equation we have derived. The key is to ensure that the equation correctly represents the relationship between the length, width, and area. One common mistake students make is to misinterpret the expression for the width. If the problem stated that the width was 2 meters longer than the length, the width would be expressed as I + 2. However, since the problem states that the width is 2 meters shorter, we correctly express it as I - 2. Another important point to consider is the order of operations. The multiplication of I and (I - 2) must be performed before equating it to the area. This is a basic algebraic principle that must be followed to ensure the equation is correctly represented. In summary, forming the equation correctly is about translating the geometric relationships into algebraic terms. The equation 168 = I × (I - 2) accurately represents the problem's conditions and sets the stage for finding the length of the rectangle. The next section will focus on solving this equation and interpreting the solution.
Solving for Length: Applying Mathematical Techniques
Once we have established the correct equation for length, the next step is to solve it to find the actual value of the length. In this case, the equation is 168 = I × (I - 2). This equation is a quadratic equation, which means it can be rearranged into the standard quadratic form: ax² + bx + c = 0. Let's rearrange our equation: 168 = I² - 2I can be rewritten as I² - 2I - 168 = 0. Now, we have a quadratic equation in the form ax² + bx + c = 0, where a = 1, b = -2, and c = -168. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. For this equation, factoring is a suitable method if we can find two numbers that multiply to -168 and add to -2. These two numbers are -14 and 12. So, we can factor the quadratic equation as follows: (I - 14)(I + 12) = 0. This factored form gives us two possible solutions for I: I - 14 = 0 or I + 12 = 0. Solving these equations, we get: I = 14 or I = -12. However, in the context of this problem, the length of a rectangle cannot be negative. Therefore, we discard the solution I = -12. This leaves us with the solution I = 14. So, the length of the rectangle is 14 meters. Now that we have found the length, it’s a good practice to verify our solution. We can do this by substituting the value of I back into the original equation and checking if it holds true. The original equation is: 168 = I × (I - 2). Substituting I = 14, we get: 168 = 14 × (14 - 2). Simplifying, we have: 168 = 14 × 12. 168 = 168. This confirms that our solution is correct. The length of the rectangle is indeed 14 meters. In summary, solving for the length involves rearranging the equation into a standard form, using an appropriate method (such as factoring) to find the solutions, and verifying the solution in the context of the problem. This process ensures that we arrive at the correct and meaningful answer. The ability to solve quadratic equations is a fundamental skill in algebra and has numerous applications in various fields, including geometry, physics, and engineering. In addition to factoring, the quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it can be easily factored. The quadratic formula is: I = (-b ± √(b² - 4ac)) / (2a). Applying this formula to our equation I² - 2I - 168 = 0 would also yield the same solutions, I = 14 and I = -12.
Conclusion: Key Takeaways and Practical Applications
In conclusion, solving geometric problems involving rectangles requires a systematic approach that combines geometric principles with algebraic techniques. The problem we addressed involved finding the equation to determine the length of a rectangle, given its area and the relationship between its length and width. The key takeaways from this exploration are several. First, it’s crucial to understand the problem statement thoroughly and identify the key information. This includes defining the variables, understanding the relationships between them, and recognizing the relevant formulas. In our case, the key formula was the area of a rectangle: Area = Length × Width. Second, forming the correct equation is essential. This involves translating the geometric relationships into algebraic terms. The equation 168 = I × (I - 2) accurately represented the problem's conditions and set the stage for finding the length of the rectangle. Third, solving the equation requires applying appropriate algebraic techniques. For quadratic equations, methods such as factoring, completing the square, and using the quadratic formula are valuable tools. In our case, factoring provided a straightforward way to find the solutions. Fourth, verifying the solution is a crucial step. This ensures that the answer is correct and makes sense in the context of the problem. By substituting the value of I back into the original equation, we confirmed that our solution was accurate. Finally, understanding the practical applications of these skills is important. Geometric problems are not just abstract exercises; they have real-world relevance in various fields, including architecture, engineering, and design. For instance, architects need to calculate the dimensions of rooms and buildings, and engineers need to determine the size of structures. Let's consider some practical applications of this problem-solving approach. Suppose you are designing a rectangular garden. You know the total area you want the garden to cover and have a constraint on the relationship between the length and width (e.g., the width must be a certain amount shorter than the length). The same principles we used in this problem can be applied to determine the dimensions of the garden. Another example is in manufacturing, where companies need to calculate the dimensions of materials to minimize waste and optimize production. Understanding how to set up and solve equations involving areas and dimensions is a valuable skill in these contexts. In addition to practical applications, solving geometric problems helps develop critical thinking and problem-solving skills. These skills are transferable and can be applied in various other areas of life. The ability to break down a complex problem into smaller, manageable steps, set up equations, and solve them is a valuable asset in many professions. In summary, mastering geometric problems involves a combination of geometric understanding, algebraic skills, and logical reasoning. By following a systematic approach and practicing regularly, you can develop these skills and become a proficient problem solver. The next step in advancing your understanding might involve exploring more complex geometric shapes and their properties, such as triangles, circles, and three-dimensional figures.
Solving Rectangle Length Equation in Mathematics
Solving rectangle length equations is a fundamental skill in mathematics, with applications spanning various fields from geometry to real-world problem-solving. In this article, we've explored a specific problem: a rectangle with a width 2 meters shorter than its length (I), and an area of 168 square meters. Our aim was to identify the correct equation to determine the length of this rectangle. The key to success in such problems lies in understanding the relationships between the dimensions of a rectangle and its area. The area of a rectangle is calculated by multiplying its length by its width. When presented with a problem where the width is described in terms of the length, we can express the width as a function of the length. This allows us to set up an equation using the given area. In our case, the width was 2 meters shorter than the length I, so we expressed the width as I - 2. Given the area of 168 square meters, we formed the equation: 168 = I × (I - 2). This equation accurately represents the geometric constraints of the problem. The left side represents the given area, and the right side represents the product of the length and width. The next crucial step is to analyze and solve this equation. Recognizing that it is a quadratic equation is essential. We can expand and rearrange it into the standard quadratic form: I² - 2I - 168 = 0. This standard form allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Factoring, in this case, proved to be a straightforward method. By identifying the factors -14 and 12, we expressed the equation as (I - 14)(I + 12) = 0. This yielded two potential solutions: I = 14 and I = -12. However, in the context of geometric problems, we must consider the practical implications of our solutions. Length cannot be negative, so we discarded I = -12 and accepted I = 14 as the length of the rectangle. Verifying our solution is an important step to ensure accuracy. Substituting I = 14 back into the original equation, we confirmed that the area was indeed 168 square meters. This verification step provides confidence in our answer and highlights the importance of checking our work. The problem we solved is a classic example of how algebra and geometry intersect. It demonstrates how algebraic equations can be used to model and solve geometric problems. The ability to translate word problems into mathematical equations is a fundamental skill in mathematics, with applications in various fields. Beyond this specific problem, the techniques we've discussed are applicable to a wide range of geometric problems involving rectangles and other shapes. Understanding the properties of geometric shapes and their relationships allows us to set up equations and solve for unknown dimensions. In summary, solving rectangle length equations involves understanding geometric principles, forming algebraic equations, applying appropriate solving techniques, and verifying solutions. These skills are essential for success in mathematics and for solving real-world problems involving geometric shapes and dimensions.
