Problem 118 - A Graphical Approach to College Algebra

Question

Error Tolerances Suppose that a square picture frame has sides that vary between 9.9 inches and 10.1 inches. What range of values is possible for the perimeter \(P\) of the picture frame? Express your answer by using a threepart inequality.

Step-by-Step Solution

Answer

The perimeter of the picture frame ranges from 39.6 to 40.4 inches.

Step 1: Understand the Perimeter Formula

The perimeter of a square is given by the formula \( P = 4s \), where \( s \) is the length of one side of the square. Here, since the side length varies, we need to consider the range of values for \( s \).

Step 2: Identify Side Length Range

We know the side length of the picture frame varies between 9.9 inches and 10.1 inches. So the side length \( s \) satisfies the inequality \( 9.9 \leq s \leq 10.1 \).

Step 3: Calculate Minimum Perimeter

Using the smallest possible side length, 9.9 inches, calculate the minimum perimeter by substituting \( s = 9.9 \) into the perimeter formula: \[ P = 4 \times 9.9 = 39.6 \].

Step 4: Calculate Maximum Perimeter

Using the largest possible side length, 10.1 inches, calculate the maximum perimeter by substituting \( s = 10.1 \) into the perimeter formula: \[ P = 4 \times 10.1 = 40.4 \].

Step 5: Formulate the Three-Part Inequality

Now, use the calculated minimum and maximum perimeters to express the range of perimeters as a three-part inequality: \( 39.6 \leq P \leq 40.4 \).

Key Concepts

Square Geometry

Square geometry is a fascinating topic in mathematics, especially when it comes to understanding its properties and calculations. A square is a special type of quadrilateral with four equal sides and four right angles. This simplicity allows us to easily calculate many aspects of the square, such as the perimeter and area.

When we talk about the perimeter of a square, it refers to the total length around the square. In simpler terms, you can find it by adding up all the sides of the square. Since all the sides are equal, the formula for calculating the perimeter (\( \text{P}\)) is straightforward: \[\text{P} = 4s\]where \( \text{s}\) represents the length of one side.

• A square's unique equal-sided feature streamlines these calculations.
• Its formula helps solve fun problems involving geometry in practical scenarios, like finding the material needed for frames.

Understanding square geometry lays the foundation for a wide array of geometric concepts, making it a valuable area of study.

Error Tolerance

Error tolerance is a critical concept, especially in fields that require precise measurements and manufacturing. When dealing with physical objects, it's important to understand that measurements aren't always perfect due to manufacturing limitations or measuring device inaccuracies. Hence, we introduce error tolerance to account for these deviations.

In the context of our original exercise, the side length of the square picture frame ranges from 9.9 inches to 10.1 inches. This variability illustrates the practical aspect of error tolerance. It accounts for small potential differences in dimensions and produces the range of the perimeter accordingly.

• Error tolerance allows flexibility in designs and blueprints, accommodating minor discrepancies.
• Specific ranges ensure product quality while maintaining feasible manufacturing processes.

Error tolerance ensures reliability and efficiency in real-world applications, thus making it an indispensable part of product design and quality assurance.

Inequalities

Inequalities are expressions that ascertain the relation between two values, indicating that one value is either greater than, less than, or equal to another. They come in handy when aiming to describe varied scenarios, like ranges or limits. Inequalities are prevalent in several mathematical contexts and are crucial for describing situations with ranges or conditions.

Considering our example of the square frame, we use inequalities to express the range of possible perimeter values. We start with the known range of side lengths, \(9.9 \leq s \leq 10.1\), and transform it through the perimeter formula. This yields: \[39.6 \leq P \leq 40.4\]This three-part inequality is insightful because it neatly encapsulates all the possible perimeter values based on the side length's variability.

• Inequalities help effectively communicate the possible outcomes or constraints.
• Mastering inequalities can aid in solving tasks involving unknown variables and constraints.

Being able to set and interpret inequalities is fundamental in algebra and calculus, proving essential for mathematical problem-solving.