Solving for the Greatest and Least Acceptable Values

Oct 13, 2025 by windailys.com 146 views

Introduction

In the manufacturing industry, quality control is a crucial aspect to ensure that products meet the required standards. One such example is the thickness of a rubber shoe sole, which must be within a specific range for the shoe to pass inspection. In this article, we will discuss how to find the greatest and least acceptable values of the rubber shoe sole thickness using mathematical methods.

Problem Statement

A rubber shoe sole is specified to be 1.75 inches thick. The actual thickness of a rubber shoe sole must be within 0.02 inch of the specified thickness for the shoe to pass inspection. This means that the acceptable thickness range is between 1.75 - 0.02 and 1.75 + 0.02 inches.

Mathematical Formulation

Let's denote the specified thickness as t_spec and the acceptable thickness range as t_accept. We can write the mathematical equation for the acceptable thickness range as:

t_accept = [t_spec - 0.02, t_spec + 0.02]

where t_spec is the specified thickness and 0.02 is the acceptable deviation.

Solving for the Greatest and Least Acceptable Values

To find the greatest and least acceptable values, we need to solve the equation for t_accept. We can do this by substituting the value of t_spec into the equation:

t_accept = [1.75 - 0.02, 1.75 + 0.02]

Simplifying the equation, we get:

t_accept = [1.73, 1.77]

Therefore, the greatest acceptable value is 1.77 inches and the least acceptable value is 1.73 inches.

Conclusion

In this article, we discussed how to find the greatest and least acceptable values of the rubber shoe sole thickness using mathematical methods. By solving the equation for the acceptable thickness range, we found that the greatest acceptable value is 1.77 inches and the least acceptable value is 1.73 inches. This demonstrates the importance of quality control in the manufacturing industry and the need for precise calculations to ensure that products meet the required standards.

Acceptable Thickness Range

The acceptable thickness range for the rubber shoe sole is between 1.73 and 1.77 inches.

Greatest Acceptable Value

The greatest acceptable value for the rubber shoe sole thickness is 1.77 inches.

Least Acceptable Value

The least acceptable value for the rubber shoe sole thickness is 1.73 inches.

Mathematical Derivation

The mathematical derivation for the acceptable thickness range is as follows:

t_accept = [t_spec - 0.02, t_spec + 0.02]

Substituting the value of t_spec, we get:

t_accept = [1.75 - 0.02, 1.75 + 0.02]

Simplifying the equation, we get:

t_accept = [1.73, 1.77]

Therefore, the greatest acceptable value is 1.77 inches and the least acceptable value is 1.73 inches.

Real-World Applications

The problem of finding the greatest and least acceptable values of the rubber shoe sole thickness has real-world applications in the manufacturing industry. Quality control is a crucial aspect of ensuring that products meet the required standards, and precise calculations are necessary to ensure that products are within the acceptable thickness range.

Future Research Directions

Future research directions in this area could include:

Investigating the effects of temperature and humidity on the thickness of the rubber shoe sole
Developing new mathematical models to predict the acceptable thickness range
Conducting experiments to validate the mathematical models

Introduction

In our previous article, we discussed how to find the greatest and least acceptable values of the rubber shoe sole thickness using mathematical methods. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the specified thickness of the rubber shoe sole?

A: The specified thickness of the rubber shoe sole is 1.75 inches.

Q: What is the acceptable deviation from the specified thickness?

A: The acceptable deviation from the specified thickness is 0.02 inch.

Q: How do I find the greatest and least acceptable values of the rubber shoe sole thickness?

A: To find the greatest and least acceptable values, you can use the following equation:

t_accept = [t_spec - 0.02, t_spec + 0.02]

where t_spec is the specified thickness and 0.02 is the acceptable deviation.

Q: What are the greatest and least acceptable values of the rubber shoe sole thickness?

A: The greatest acceptable value is 1.77 inches and the least acceptable value is 1.73 inches.

Q: Why is it important to ensure that the rubber shoe sole thickness is within the acceptable range?

A: Ensuring that the rubber shoe sole thickness is within the acceptable range is important because it affects the quality and performance of the shoe. If the thickness is too thin or too thick, it can lead to problems such as reduced durability, increased wear and tear, and compromised safety.

Q: Can I use this method to find the acceptable thickness range for other types of materials?

A: Yes, you can use this method to find the acceptable thickness range for other types of materials. However, you will need to adjust the equation to account for the specific material properties and requirements.

Q: What are some common applications of this method?

A: This method is commonly used in the manufacturing industry to ensure that products meet the required standards. It is also used in quality control to monitor and improve the production process.

Q: Can I use this method to find the acceptable thickness range for a specific product?

A: Yes, you can use this method to find the acceptable thickness range for a specific product. However, you will need to gather data on the product's specifications and requirements, and then use the equation to calculate the acceptable thickness range.

Q: What are some potential limitations of this method?

A: Some potential limitations of this method include:

The equation assumes a linear relationship between the thickness and the acceptable range, which may not always be the case.
The equation does not take into account other factors that may affect the acceptable thickness range, such as temperature and humidity.
The equation may not be applicable to all types of materials or products.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the greatest and least acceptable values of the rubber shoe sole thickness. We hope that this Q&A article has provided you with a better understanding of the problem and its applications.

Frequently Asked Questions

Q: What is the specified thickness of the rubber shoe sole?
A: The specified thickness of the rubber shoe sole is 1.75 inches.
Q: What is the acceptable deviation from the specified thickness?
A: The acceptable deviation from the specified thickness is 0.02 inch.
Q: How do I find the greatest and least acceptable values of the rubber shoe sole thickness?
A: To find the greatest and least acceptable values, you can use the following equation: t_accept = [t_spec - 0.02, t_spec + 0.02]
Q: What are the greatest and least acceptable values of the rubber shoe sole thickness?
A: The greatest acceptable value is 1.77 inches and the least acceptable value is 1.73 inches.

A Rubber Shoe Sole Thickness Problem: Finding the Greatest and Least Acceptable Values
A Rubber Shoe Sole Thickness Problem: Mathematical Derivation
A Rubber Shoe Sole Thickness Problem: Real-World Applications

Glossary

Specified thickness: The thickness of the rubber shoe sole as specified by the manufacturer.
Acceptable deviation: The maximum amount by which the actual thickness can deviate from the specified thickness.
Greatest acceptable value: The maximum thickness of the rubber shoe sole that is still within the acceptable range.
Least acceptable value: The minimum thickness of the rubber shoe sole that is still within the acceptable range.