Problem 53 - 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 you found in part (a).

Step-by-Step Solution

Answer

The range of thicknesses is given by the inequality \(0.017 \leq x \leq 0.023\).

Step 1: Define the Problem

We need to find an inequality that describes the range of possible thickness for the laminate, which has a nominal thickness of 0.020 inches and a tolerance of 0.003 inches.

Step 2: Formulate the Inequality with Absolute Values

The actual thickness can deviate by at most 0.003 inches from the nominal thickness of 0.020 inches. We express this with the inequality: \[ |x - 0.020| \leq 0.003 \]where \(x\) represents the actual thickness of the laminate.

Step 3: Interpret the Absolute Value Inequality

An absolute value inequality \(|x - a| \leq b\) translates to two inequalities: \[-b \leq x-a \leq b\]. Apply this concept to our inequality.

Step 4: Solve the Inequalities

First, we break down the inequality \(|x - 0.020| \leq 0.003\) into:1. \(x - 0.020 \leq 0.003\)2. \(x - 0.020 \geq -0.003\).Solving these inequalities:1. \(x \leq 0.020 + 0.003 = 0.023\)2. \(x \geq 0.020 - 0.003 = 0.017\).

Step 5: Write the Solution as a Combined Inequality

Combine the results from Step 4 to get the final solution of the inequality:\[0.017 \leq x \leq 0.023\].This describes the range of thicknesses for the laminate under the given tolerance.

Key Concepts

Absolute Value

The concept of absolute value is essential in understanding the exercise about laminate thickness. Absolute value refers to the distance a number is from zero on the number line, regardless of direction. It's always a non-negative value. For instance, both 5 and -5 have an absolute value of 5.

In the context of this problem, absolute values help express how much a measurement can vary from a central point. When we say \( |x - 0.020| \), we are looking at how far away the actual thickness \(x\) is from the standard thickness of 0.020 inches.

• Represents deviation: The expression \( |x - 0.020| \) captures any deviations from the target thickness of 0.020 inches.
• Ensures consistency: By using absolute values, we account for both positive and negative deviations, ensuring a complete range of permissible variations.

Inequality Solving

Inequality solving is a critical skill in mathematics that helps to find a range of possible values that satisfy certain conditions. With absolute value inequalities, like the one in this exercise, it's all about understanding how far you can deviate from a predetermined point.

When faced with an absolute value inequality \( |x - a| \leq b \), it can be resolved by breaking it down into two separate inequalities that do not include absolute values:

• \( x - a \leq b \)
• \( x - a \geq -b \)

These are then solved individually to find the solution set. In our specific case, it involved determining such values of thickness \(x\) that maintain the allowable tolerance from the desired number, which results in a range of acceptable thicknesses.

Mathematical Modeling

Mathematical modeling is the practice of using mathematics to simulate real-world situations and solve practical problems. In this exercise, the model is built around the manufacturing process of laminates that must adhere to strict thickness specifications.

By constructing an inequality involving absolute values, we create a model that reflects the allowable variation in laminate thickness. Here’s how mathematical modeling benefits this specific problem:

• Simplifies complexity: Transforming a production requirement into a mathematical equation makes it easier to analyze and solve.
• Predicts allowable ranges: Modeling helps predict what the acceptable parameters are, ensuring the produced laminates meet quality standards.
• Facilitates decision-making: Provides a clear mathematical boundary within which the product specifications must lie.

Tolerance

Tolerance in manufacturing and engineering refers to the allowable deviation in a physical dimension. It's crucial for ensuring quality and functionality of products. For instance, in this exercise, tolerance defines how much the thickness of the laminate can deviate from the ideal 0.020 inches.

Understanding tolerance is key to solving the inequality provided in this exercise. Tolerance of 0.003 inches means we can either add or subtract this value from the nominal thickness to find the acceptable range. Thus, constructing and solving the inequality \( |x - 0.020| \leq 0.003 \) gives us the limits within which the laminate thickness must fall:

• Maximum thickness: 0.020 + 0.003 = 0.023 inches.
• Minimum thickness: 0.020 - 0.003 = 0.017 inches.

This range \( 0.017 \leq x \leq 0.023 \) ensures the product meets quality standards while accounting for minor variations during production.