Calculating the Tolerable Range

Oct 13, 2025 by windailys.com 110 views

Introduction

In the world of manufacturing, quality control is a crucial aspect of producing high-quality products. A company that produces square tiles is no exception. The advertised area of each tile is $100 \, \text{cm}^2$ , but there is some error in the production process. The company wants to ensure that the tiles produced differ by no more than $5\%$ from the advertised area. In this article, we will explore the mathematical concepts behind this quality control dilemma and provide a solution to ensure that the tiles meet the advertised area.

The Problem

Let's assume that the side length of the square tile is $s$ cm. The area of the tile is given by the formula $A = s^2$ . The company wants to ensure that the area of each tile differs by no more than $5\%$ from the advertised area of $100 \, \text{cm}^2$ . This means that the area of each tile should be between $95\%$ and $105\%$ of the advertised area.

Mathematically, this can be expressed as:

0.95 \times 100 \leq s^2 \leq 1.05 \times 100

Simplifying the inequality, we get:

95 \leq s^2 \leq 105

Taking the square root of both sides, we get:

\sqrt{95} \leq s \leq \sqrt{105}

Calculating the Tolerable Range

To find the tolerable range of side lengths, we need to calculate the square roots of $95$ and $105$ . Using a calculator, we get:

\sqrt{95} \approx 9.74 \, \text{cm}

\sqrt{105} \approx 10.25 \, \text{cm}

Therefore, the tolerable range of side lengths is between $9.74$ cm and $10.25$ cm.

Ensuring Quality Control

To ensure that the tiles meet the advertised area, the company can implement the following quality control measures:

Measurement: Measure the side length of each tile using a precision caliper or a micrometer.
Calculation: Calculate the area of each tile using the formula $A = s^2$ .
Verification: Verify that the area of each tile falls within the tolerable range of $95 \, \text{cm}^2$ to $105 \, \text{cm}^2$ .
Rejection: Reject any tiles that do not meet the quality control standards.

Conclusion

In conclusion, ensuring that square tiles meet the advertised area is a complex problem that requires careful consideration of the mathematical concepts involved. By implementing the quality control measures outlined above, the company can ensure that the tiles produced differ by no more than $5\%$ from the advertised area.

Future Directions

In the future, the company may want to consider implementing more advanced quality control measures, such as:

Statistical Process Control: Use statistical methods to monitor and control the production process.
Machine Learning: Use machine learning algorithms to predict the quality of the tiles based on various factors, such as the side length and the production process.
Automated Inspection: Use automated inspection systems to quickly and accurately inspect the tiles for quality.

By implementing these advanced quality control measures, the company can further improve the quality of its products and maintain its competitive edge in the market.

References

[1] "Quality Control in Manufacturing" by John Wiley & Sons
[2] "Statistical Process Control" by ASQ
[3] "Machine Learning for Quality Control" by Springer

Appendix

The following is a list of mathematical formulas and equations used in this article:

$A = s^2$ (area of a square tile)
$0.95 \times 100 \leq s^2 \leq 1.05 \times 100$ (inequality for the tolerable range of side lengths)
$\sqrt{95} \leq s \leq \sqrt{105}$ (inequality for the tolerable range of side lengths)

Introduction

In our previous article, we explored the mathematical concepts behind ensuring that square tiles meet the advertised area. We discussed the importance of quality control in manufacturing and provided a solution to ensure that the tiles produced differ by no more than $5\%$ from the advertised area. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q&A

Q: What is the main goal of quality control in manufacturing?

A: The main goal of quality control in manufacturing is to ensure that the products produced meet the required standards and specifications.

Q: Why is it important to ensure that the tiles meet the advertised area?

A: It is essential to ensure that the tiles meet the advertised area because it affects the customer's satisfaction and the company's reputation. If the tiles do not meet the advertised area, it can lead to customer complaints and financial losses.

Q: How can the company ensure that the tiles meet the advertised area?

A: The company can ensure that the tiles meet the advertised area by implementing quality control measures such as measurement, calculation, verification, and rejection.

Q: What is the tolerable range of side lengths for the tiles?

A: The tolerable range of side lengths for the tiles is between $9.74$ cm and $10.25$ cm.

Q: How can the company implement statistical process control to monitor and control the production process?

A: The company can implement statistical process control by collecting data on the production process, analyzing the data, and making adjustments as needed.

Q: What is machine learning, and how can it be used in quality control?

A: Machine learning is a type of artificial intelligence that can be used to predict the quality of the tiles based on various factors such as the side length and the production process.

Q: How can the company use automated inspection systems to quickly and accurately inspect the tiles for quality?

A: The company can use automated inspection systems such as computer vision and robotics to quickly and accurately inspect the tiles for quality.

Q: What are some common mistakes that companies make when implementing quality control measures?

A: Some common mistakes that companies make when implementing quality control measures include:

Not setting clear quality standards
Not training employees on quality control procedures
Not regularly inspecting and testing products
Not addressing quality control issues promptly

Q: How can the company ensure that the quality control measures are effective?

A: The company can ensure that the quality control measures are effective by regularly monitoring and evaluating the quality of the products, making adjustments as needed, and providing ongoing training and education to employees.

Conclusion

In conclusion, ensuring that square tiles meet the advertised area is a complex problem that requires careful consideration of the mathematical concepts involved. By implementing quality control measures and addressing common mistakes, companies can ensure that their products meet the required standards and specifications.