Inequality For Perimeter Of Rectangle Problem Solution
This article delves into the process of identifying the appropriate inequality to represent the constraints on the perimeter of a rectangular steel sheet. The problem involves a rectangular sheet where the length is four times its width, and the perimeter must be less than 100 inches. We will explore the steps involved in translating this word problem into a mathematical inequality, providing a comprehensive explanation to ensure clarity and understanding.
Problem Statement Breakdown
In mathematical problem-solving, the initial step is to meticulously break down the problem statement. Our problem presents a scenario involving a rectangular sheet of steel. Two key dimensions define a rectangle: its length and its width. The problem explicitly states a relationship between these dimensions: the length (l) is four times the width (w). This relationship is crucial and can be expressed mathematically as l = 4w. The problem further introduces a constraint on the perimeter of the sheet. The perimeter, which is the total distance around the rectangle, must be less than 100 inches. This constraint is the foundation for formulating our inequality.
Understanding the relationship between the length and width is paramount. It allows us to express both dimensions in terms of a single variable, simplifying the process of calculating the perimeter. The perimeter of a rectangle is generally calculated as P = 2l + 2w, where P represents the perimeter, l the length, and w the width. In our case, since l = 4w, we can substitute 4w for l in the perimeter formula. This substitution yields P = 2(4w) + 2w, which simplifies to P = 8w + 2w, and further to P = 10w. This simplified expression is a significant step towards formulating the inequality.
The constraint on the perimeter, that it must be less than 100 inches, is a critical piece of information. In mathematical terms, “less than” is represented by the inequality symbol “<”. Therefore, we know that the perimeter, which we have expressed as 10w, must be less than 100. This directly translates to the inequality 10w < 100. This inequality is the core of our problem and represents the mathematical expression of the given constraints. It states that ten times the width of the rectangular sheet must be less than 100 inches to satisfy the given conditions.
Formulating the Inequality
To formulate the correct inequality, we need to translate the given information into a mathematical statement. The key pieces of information are the relationship between the length and width (l = 4w) and the constraint on the perimeter (less than 100 inches). As previously discussed, the perimeter of a rectangle is given by P = 2l + 2w. Substituting l = 4w into this formula, we get P = 2(4w) + 2w, which simplifies to P = 10w. The constraint that the perimeter must be less than 100 inches is expressed as P < 100. Now, we substitute our expression for P, which is 10w, into this inequality, resulting in 10w < 100. This inequality represents the condition that ten times the width must be less than 100 inches.
The inequality 10w < 100 is a linear inequality in one variable, w. It represents a range of possible values for the width that satisfy the given conditions. To fully understand the implications of this inequality, it is often helpful to solve it for w. Dividing both sides of the inequality by 10, we get w < 10. This tells us that the width of the rectangular sheet must be less than 10 inches. This result provides a clear and concise understanding of the limitation on the width imposed by the perimeter constraint.
The process of formulating this inequality highlights the importance of translating word problems into mathematical expressions. It requires careful identification of the given relationships and constraints, and the ability to represent these using mathematical symbols and formulas. In this case, understanding the formula for the perimeter of a rectangle and the meaning of the “less than” inequality were crucial steps in arriving at the correct inequality. The final inequality, 10w < 100, accurately captures the condition that the perimeter of the rectangular sheet, with the given length-to-width ratio, must be less than 100 inches.
Solving and Interpreting the Inequality
Solving the inequality 10w < 100 involves isolating the variable w to determine the range of possible values for the width. As mentioned earlier, we divide both sides of the inequality by 10. Since 10 is a positive number, the direction of the inequality remains unchanged. This gives us w < 10. This solution signifies that the width (w) of the rectangular sheet must be less than 10 inches to satisfy the given conditions.
Interpreting the solution w < 10 in the context of the problem is crucial. It means that any width value less than 10 inches will result in a perimeter less than 100 inches, given that the length is four times the width. For instance, if the width is 5 inches, the length would be 4 * 5 = 20 inches, and the perimeter would be 2 * 5 + 2 * 20 = 50 inches, which is less than 100 inches. Similarly, if the width is 9 inches, the length would be 4 * 9 = 36 inches, and the perimeter would be 2 * 9 + 2 * 36 = 90 inches, also less than 100 inches. However, if the width is 10 inches or more, the perimeter would exceed 100 inches, violating the problem's constraint.
The solution w < 10 also implies a practical limitation on the dimensions of the steel sheet. In a real-world scenario, knowing that the width cannot exceed 10 inches is vital for material selection and cutting processes. It allows for efficient planning and minimizes material wastage. Furthermore, understanding the range of possible widths helps in determining the range of possible lengths, as the length is directly dependent on the width (l = 4w). If w < 10, then l < 40. This means the length must be less than 40 inches.
The process of solving and interpreting inequalities is a fundamental skill in mathematics and has wide-ranging applications in various fields. In this specific problem, it allows us to determine the allowable dimensions of a rectangular sheet given a constraint on its perimeter. The solution provides not only a mathematical answer but also practical insights into the real-world implications of the problem.
Common Mistakes to Avoid
When working with inequalities, there are several common mistakes that students and problem-solvers often make. Recognizing and avoiding these pitfalls is crucial for accurate problem-solving. One common mistake is incorrectly applying the inequality symbols. For example, confusing “less than” (<) with “less than or equal to” (≤) or misinterpreting “greater than” (>) as “less than.” In our problem, the perimeter must be strictly less than 100 inches, so using “≤” instead of “<” would be incorrect.
Another frequent error is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This rule is a fundamental property of inequalities and is essential for maintaining the correctness of the solution. In our problem, we only divided by a positive number (10), so this was not an issue. However, in other problems, this can lead to a completely wrong solution.
A third mistake is not correctly translating the word problem into a mathematical expression. This involves carefully identifying the relationships and constraints described in the problem statement and representing them using appropriate variables, symbols, and equations or inequalities. In our case, a misunderstanding of the perimeter formula or the relationship between length and width could lead to an incorrect inequality.
Furthermore, students sometimes make errors in the algebraic manipulation of inequalities. This can involve incorrect simplification, combining like terms, or applying the distributive property. It’s important to meticulously check each step in the algebraic process to avoid such errors. Another common oversight is not interpreting the solution in the context of the problem. While solving the inequality is a crucial step, it is equally important to understand what the solution means in the real-world scenario. For instance, in our problem, the solution w < 10 means that the width must be less than 10 inches. This interpretation provides practical information about the dimensions of the steel sheet.
Finally, neglecting to check the solution is a mistake that can easily be avoided. Substituting the solution back into the original inequality can verify its correctness. If the solution does not satisfy the inequality, it indicates an error in the problem-solving process. By being mindful of these common mistakes, problem-solvers can significantly improve their accuracy and understanding when working with inequalities.
Conclusion
In conclusion, selecting the correct inequality to represent a given problem requires a thorough understanding of the problem statement, careful translation of the information into mathematical terms, and attention to the properties of inequalities. In the case of the rectangular steel sheet, we successfully formulated the inequality 10w < 100 by recognizing the relationship between the length and width, applying the perimeter formula, and incorporating the constraint on the perimeter. This inequality accurately captures the condition that the perimeter must be less than 100 inches.
The process involved breaking down the problem into smaller, manageable parts, identifying the key variables and relationships, and expressing these mathematically. We then solved the inequality to determine the range of possible values for the width, w < 10. This solution provides a clear and concise understanding of the limitations on the width imposed by the perimeter constraint. Furthermore, we discussed the importance of interpreting the solution in the context of the problem, which provides practical insights into the dimensions of the steel sheet.
Throughout this exploration, we highlighted common mistakes to avoid when working with inequalities, such as misinterpreting inequality symbols, neglecting to reverse the inequality sign when multiplying or dividing by a negative number, and incorrectly translating word problems into mathematical expressions. By being aware of these potential pitfalls, problem-solvers can enhance their accuracy and problem-solving skills.
The ability to formulate and solve inequalities is a fundamental skill in mathematics with wide-ranging applications in various fields, including engineering, economics, and computer science. Mastering this skill requires practice, attention to detail, and a solid understanding of the underlying concepts. By carefully following the steps outlined in this article, one can confidently select the correct inequality to represent a given problem and arrive at an accurate solution. The process not only provides a mathematical answer but also enhances problem-solving abilities and critical thinking skills, which are essential in both academic and real-world contexts.
