Problem 100 - A Graphical Approach to Precalculus with Limits

Question

An aluminum can is designed to have a height \(H\) of 5 inches with an error tolerance of not more than 0.05 inch. Write an absolute value inequality that describes all values of \(H\) that satisfy this restriction.

Step-by-Step Solution

Answer

\(|H - 5| \leq 0.05\)

Step 1: Define the Error Tolerance

The height of the can is designed to be 5 inches, but there is an allowable deviation of \(0.05\) inch. This means that any deviation should not exceed \(0.05\) inch.

Step 2: Express Tolerance using Inequality

The height \(H\) can vary up or down by no more than \(0.05\) inches from 5 inches. This can be expressed as an inequality: \(|H - 5| \leq 0.05\). This inequality states that the absolute difference between the height \(H\) and the desired height 5 inches must be less than or equal to \(0.05\) inches.

Step 3: Interpret the Inequality

The absolute value inequality \(|H - 5| \leq 0.05\) means the height \(H\) must be between 4.95 inches and 5.05 inches. This is derived from breaking down the inequality to two separate inequalities: \(H - 5 \leq 0.05\) and \(5 - H \leq 0.05\). Both simplify to \(4.95 \leq H \leq 5.05\).

Key Concepts

Error Tolerance

When discussing measurements, such as the height of an aluminum can, error tolerance is crucial. It provides the limits within which the actual measurements can vary from the desired or intended value. Imagine the can is meant to be exactly 5 inches high, but due to manufacturing processes, achieving precise measurements every time is challenging. Error tolerance allows for this slight deviation and ensures that even with a small variation, the product remains acceptable.
For instance, if the error tolerance is 0.05 inches, the actual height can differ slightly up or down within this range. This is a very useful concept in quality control and manufacturing where precise measurements are critical.

• Error tolerance defines the range of acceptable deviation.
• Ensures products meet quality standards despite minor variations.
• Provides flexibility in production processes.

Inequality

An inequality is a mathematical statement containing symbols like <, >, ≤, or ≥. It shows the relationship between two values where one is not necessarily equal to the other. In the case of the aluminum can, we are interested in the heights that satisfy the error tolerance condition. This is captured using an inequality.
The inequality in question is \(|H - 5| \leq 0.05\). This indicates that the deviation from the target height of 5 inches should not be more than 0.05 inches.

• Inequalities express a range of possible values.
• They can show bounds within which a variable must lie.
• Help in defining conditions for scenarios like error tolerances.

Absolute Value

Absolute value refers to the distance a number is from zero on the number line, without considering direction. This means absolute values are always positive. When we say \(|H - 5| \), we're referring to how much the actual height varies from the target height of 5 inches, regardless of whether it's greater or lesser.
This concept is particularly useful in real-world problems like error tolerance because it allows us to measure the deviation or error, capturing both positive and negative discrepancies.

• Absolute value focuses on magnitude, ignoring direction.
• Helps in quantifying both upward and downward deviations.
• Makes expressions of error and variation more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the building block for defining mathematical relationships and solving equations. In our case, we have the expression \(H - 5\) within the inequality, which represents the deviation of the can's height from its ideal.
By wrapping this expression in an absolute value and setting it in an inequality, we create a definitive mathematical model that describes an acceptable range of heights.
Understanding algebraic expressions is key to manipulating and rearranging equations to find solutions.

• Algebraic expressions combine numbers and variables.
• Used to create equations and inequalities.
• Essential for modeling and solving real-world problems.