Problem 57 - Algebra and Trigonometry

Question

Thickness of a Laminate A company manufactures industrial laminates (thin nylon-based sheets) of thickness 0.020 in, with a tolerance of 0.003 in. (a) Find an inequality involving absolute values that describes the range of possible thickness for the laminate. (b) Solve the inequality that you found in part (a).

Step-by-Step Solution

Answer

The possible thickness range is \(0.017 \leq x \leq 0.023\).

Step 1: Understanding the Problem

The problem is about finding the range of possible thicknesses for a laminate, given a standard thickness and a tolerance. The tolerance tells us how much the actual thickness can deviate from this standard.

Step 2: Creating an Inequality with Absolute Values

Let's represent the thickness by the variable \( x \). The standard thickness is 0.020 inches (in), and the tolerance is 0.003 in. The thickness can deviate from 0.020 in by as much as the tolerance. This can be expressed as an absolute value inequality: \(|x - 0.020| \leq 0.003\).

Step 3: Solving the Absolute Value Inequality

To solve the inequality \(|x - 0.020| \leq 0.003\), we split it into two separate inequalities: \(x - 0.020 \leq 0.003\) and \(x - 0.020 \geq -0.003\).

Step 4: Solving the First Inequality

Solve \(x - 0.020 \leq 0.003\) by adding 0.020 to both sides: \(x \leq 0.023\).

Step 5: Solving the Second Inequality

Solve \(x - 0.020 \geq -0.003\) by adding 0.020 to both sides: \(x \geq 0.017\).

Step 6: Writing the Final Solution

Combine the results from the two inequalities to express the range of thickness: \(0.017 \leq x \leq 0.023\).

Key Concepts

Industrial Laminate

Industrial laminate refers to thin sheets, often made from materials like nylon, that are used in a wide array of applications due to their durability and lightweight nature. These sheets are particularly favored in industries that require materials resistant to wear and tear. Manufacturers of industrial laminates need to maintain strict quality standards to ensure that each sheet meets the specifications required for its end use. This involves regulating not only the components that make up the laminate but also its dimensions, particularly thickness. When discussing laminates, allowing a slight variance is crucial, as it can impact the performance and fit of the sheets in their final application. The need for precise thickness comes from the requirement for laminates to fit perfectly into specified spaces or to work harmoniously with other components. Manufacturers accomplish this by setting a standard thickness, accompanied by a defined tolerance level, as a benchmark to ensure consistency in production.

Thickness Variation

Thickness variation in industrial laminates refers to the phenomenon where manufactured sheets are not all identical in thickness. This can happen due to different stages of the manufacturing process or the type of material used. For this reason, manufacturers set a specified 'standard thickness' that all laminates are supposed to achieve.Standard thickness can vary depending on the intended use of the laminate. In our example, the company set a thickness of 0.020 inches as the target. Any deviation from this standard is accounted for by specifying a tolerance. This tolerance allows for slight changes in thickness without affecting the overall usability or quality of the laminate.To understand and measure thickness variation, manufacturers frequently use the concept of absolute value inequalities. Absolute value inequalities provide a mathematical way to express and calculate permissible deviations from the set thickness standard. For instance, a tolerance of 0.003 inches would imply that the actual thickness can range slightly above or below the specified measurement, captured by the inequality formula \( |x - 0.020| \leq 0.003 \).

Tolerance

Tolerance in this context describes the permissible limit or limits of variation in the thickness of an industrial laminate. It is a critical factor in manufacturing because it establishes how much deviation from a nominal thickness is acceptable. Tolerance ensures functional conformity while still allowing for small, unavoidable variances that occur during production.By defining a tolerance, manufacturers set the boundaries within which all laminates must fall to guarantee satisfactory performance. Tolerance is expressed as an absolute value inequality, specifying the allowable range of thickness.In our example, the tolerance is set at 0.003 inches. This means the thickness of the laminate can vary between 0.017 inches and 0.023 inches, as illustrated in the solved inequality \( 0.017 \leq x \leq 0.023 \). Tolerance is essential because it helps maintain the structural integrity and function of industrial laminates. It guarantees that all products are within an acceptable range and prevents excess waste, time, and cost overruns associated with materials that fall outside the specified limits.