The Validity of Negative Values

Oct 13, 2025 by recoveryshake.com 92 views

When Solving the Application Problem, "A Rectangular Picture is Twice as Long as it is Wide. The Area of the Picture is 488 Square Inches," Solutions with Negative Values Would Not Make Sense for This Problem.

True or False: A Critical Analysis of the Problem

When it comes to solving application problems in mathematics, it's essential to consider the context and the constraints of the problem. In this case, we're given a rectangular picture that is twice as long as it is wide, with an area of 488 square inches. The question asks whether solutions with negative values would make sense for this problem. In this article, we'll delve into the details of the problem, analyze the possible solutions, and determine whether negative values are a viable option.

Understanding the Problem

Let's start by breaking down the problem and understanding what's being asked. We're given a rectangular picture with an area of 488 square inches. The picture is also described as being twice as long as it is wide. This means that if we let the width of the picture be x, then the length of the picture would be 2x.

The Area of a Rectangle

The area of a rectangle is given by the formula A = length × width. In this case, the area is 488 square inches, and the length is 2x, while the width is x. So, we can set up the equation:

2x × x = 488

Simplifying the equation, we get:

2x^2 = 488

Solving for x

To solve for x, we need to isolate the variable x. We can do this by dividing both sides of the equation by 2:

x^2 = 488/2

x^2 = 244

Now, we need to take the square root of both sides of the equation to solve for x:

x = √244

x = 15.62 (approximately)

The Length of the Picture

Now that we have the value of x, we can find the length of the picture by multiplying the width by 2:

Length = 2x = 2 × 15.62 = 31.24 (approximately)

The Validity of Negative Values

The question asks whether solutions with negative values would make sense for this problem. In this case, the width and length of the picture cannot be negative, as they represent physical dimensions. Therefore, any solution that results in a negative value for the width or length is not valid.

Conclusion

The Importance of Context in Mathematics

This problem highlights the importance of considering the context and constraints of a problem in mathematics. When solving application problems, it's essential to think about the real-world implications of the solution and ensure that it makes sense in the context of the problem. In this case, the problem requires a positive solution for the width and length of the picture, as negative values would not be physically meaningful.

Real-World Applications of Mathematics

Mathematics is often used to model real-world problems and phenomena. In this case, the problem of finding the dimensions of a rectangular picture is a classic example of a mathematical application. By solving this problem, we can gain a deeper understanding of the mathematical concepts involved and develop problem-solving skills that can be applied to a wide range of real-world situations.

The Role of Mathematics in Problem-Solving

Mathematics plays a critical role in problem-solving, as it provides a powerful tool for analyzing and understanding complex problems. By using mathematical concepts and techniques, we can break down complex problems into manageable parts, identify patterns and relationships, and develop creative solutions. In this case, the problem of finding the dimensions of a rectangular picture requires the application of mathematical concepts such as algebra and geometry.

The Benefits of Mathematics Education

Mathematics education is essential for developing problem-solving skills, critical thinking, and analytical reasoning. By studying mathematics, students can develop a deeper understanding of mathematical concepts and techniques, which can be applied to a wide range of real-world situations. In this case, the problem of finding the dimensions of a rectangular picture is a classic example of a mathematical application that requires the application of algebra and geometry.

Conclusion

In conclusion, solutions with negative values would not make sense for this problem. The width and length of the picture must be positive, as they represent physical dimensions. The correct solution to the problem is x = 15.62 (approximately), which results in a positive value for the width and length of the picture. This problem highlights the importance of considering the context and constraints of a problem in mathematics and the role of mathematics in problem-solving.

References

[1] "Algebra and Geometry" by Michael Artin
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Joy of Mathematics" by Alfred S. Posamentier

Additional Resources

[1] Khan Academy: Algebra and Geometry
[2] MIT OpenCourseWare: Mathematics
[3] Wolfram Alpha: Mathematics and Problem-Solving
Q&A: When Solving the Application Problem, "A Rectangular Picture is Twice as Long as it is Wide. The Area of the Picture is 488 Square Inches," Solutions with Negative Values Would Not Make Sense for This Problem.

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions related to the problem of finding the dimensions of a rectangular picture that is twice as long as it is wide, with an area of 488 square inches.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is A = length × width.

Q: How do we find the dimensions of the picture if it is twice as long as it is wide?

A: Let's say the width of the picture is x. Then, the length of the picture would be 2x, since it is twice as long as it is wide.

Q: How do we set up the equation to solve for x?

A: We can set up the equation by using the formula for the area of a rectangle: A = length × width. In this case, the area is 488 square inches, and the length is 2x, while the width is x. So, we can set up the equation:

2x × x = 488

Q: How do we solve for x?

A: To solve for x, we need to isolate the variable x. We can do this by dividing both sides of the equation by 2:

x^2 = 488/2

x^2 = 244

Now, we need to take the square root of both sides of the equation to solve for x:

x = √244

x = 15.62 (approximately)

Q: What is the length of the picture?

A: Now that we have the value of x, we can find the length of the picture by multiplying the width by 2:

Length = 2x = 2 × 15.62 = 31.24 (approximately)

Q: Why can't we have negative values for the width and length of the picture?

A: The width and length of the picture cannot be negative, as they represent physical dimensions. Therefore, any solution that results in a negative value for the width or length is not valid.

Q: What is the importance of considering the context and constraints of a problem in mathematics?

A: When solving application problems, it's essential to think about the real-world implications of the solution and ensure that it makes sense in the context of the problem. In this case, the problem requires a positive solution for the width and length of the picture, as negative values would not be physically meaningful.

Q: How does mathematics play a role in problem-solving?

A: Mathematics provides a powerful tool for analyzing and understanding complex problems. By using mathematical concepts and techniques, we can break down complex problems into manageable parts, identify patterns and relationships, and develop creative solutions.

Q: What are the benefits of mathematics education?

A: Mathematics education is essential for developing problem-solving skills, critical thinking, and analytical reasoning. By studying mathematics, students can develop a deeper understanding of mathematical concepts and techniques, which can be applied to a wide range of real-world situations.

Q: What are some real-world applications of mathematics?

A: Mathematics is used to model real-world problems and phenomena. Some examples of real-world applications of mathematics include:

Physics and engineering
Computer science and programming
Economics and finance
Biology and medicine

Q: How can I practice solving application problems like this one?

A: You can practice solving application problems like this one by:

Working through example problems in your textbook or online resources
Practicing with real-world applications of mathematics
Joining a study group or working with a tutor to get help and feedback

Conclusion

In conclusion, solutions with negative values would not make sense for this problem. The width and length of the picture must be positive, as they represent physical dimensions. The correct solution to the problem is x = 15.62 (approximately), which results in a positive value for the width and length of the picture. By understanding the context and constraints of the problem, we can develop problem-solving skills and apply mathematical concepts to real-world situations.

References

[1] "Algebra and Geometry" by Michael Artin
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Joy of Mathematics" by Alfred S. Posamentier

Additional Resources

[1] Khan Academy: Algebra and Geometry
[2] MIT OpenCourseWare: Mathematics
[3] Wolfram Alpha: Mathematics and Problem-Solving