Problem 96 - College Algebra with Modeling and Visualization

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 three-part inequality.

Step-by-Step Solution

Answer

The perimeter range is given by the inequality: \( 39.6 \leq P \leq 40.4 \).

Step 1: Understand the Problem

The problem asks us to find the range of possible perimeters for a picture frame with sides varying between 9.9 inches and 10.1 inches. A square perimeter is calculated as four times the side length.

Step 2: Calculate the Minimum Perimeter

To find the minimum possible perimeter, use the smallest side length. The side length can be as small as 9.9 inches, so calculate the perimeter as: \( P_{min} = 4 \times 9.9 = 39.6 \text{ inches} \).

Step 3: Calculate the Maximum Perimeter

To find the maximum possible perimeter, use the largest side length. The side length can be as large as 10.1 inches, so calculate the perimeter as: \( P_{max} = 4 \times 10.1 = 40.4 \text{ inches} \).

Step 4: Write the Range of Perimeters

The possible perimeter values range from the minimum to the maximum calculated. This can be expressed as a three-part inequality: \( 39.6 \leq P \leq 40.4 \).

Step 5: Conclusion

The perimeter of the square picture frame can vary between 39.6 inches and 40.4 inches depending on the side length variation.

Key Concepts

Error Tolerances

Error tolerances refer to the acceptable range of variation in a measurement or calculation. When measuring objects in the real world, it is common for small errors or discrepancies to occur due to limitations in measuring instruments or human error.
In this exercise, the error tolerance is clearly defined as the variation between 9.9 inches and 10.1 inches for the sides of the square picture frame.
This means that the true length of the sides could be as small as 9.9 inches or as large as 10.1 inches.

Understanding error tolerances helps us anticipate the possible range of values for measurements that are prone to variation:

• It recognizes that measurements are not always exact due to various factors.
• It helps in calculating the range within which the true measurement might fall.

Error tolerances are crucial in fields that require precision, such as engineering and construction, where they ensure safety and functionality within acceptable limits.

Perimeter Calculation

The perimeter of a shape is the total distance around the shape. For a square, the perimeter is calculated as four times the length of one side because all sides are equal. In mathematical terms, for a square with side length **s**, the perimeter **P** is calculated as:

\[ P = 4 imes s \]

In the context of this exercise, the side lengths can range from 9.9 inches to 10.1 inches, due to the established error tolerance.
Let's go through how we determine the perimeter with these varying side lengths:

• Minimum Perimeter: When the side length is at its minimum of 9.9 inches, the perimeter is \( 39.6 \) inches, calculated using \( P = 4 imes 9.9 \).
• Maximum Perimeter: Conversely, if the side length is at its maximum of 10.1 inches, the perimeter is \( 40.4 \) inches, calculated using \( P = 4 imes 10.1 \).

This calculation highlights how the perimeter can change based on side length variations due to error tolerances.

Three-Part Inequality

A three-part inequality is a concise way to express that a variable lies between two values. It is a tool used in algebra to efficiently describe the range of solutions or possibilities for a particular variable.
For example, when we say that a perimeter **P** satisfies the inequality

\[ 39.6 \leq P \leq 40.4 \]

we are indicating that **P** is greater than or equal to 39.6 inches and less than or equal to 40.4 inches.
This format clearly communicates the entire range of possible perimeters for the picture frame.

• The lower bound (39.6) ensures the perimeter isn't underestimated.
• The upper bound (40.4) ensures the perimeter isn't overestimated.

Three-part inequalities are particularly useful in solving problems where a solution must fall within a limited range, providing clarity and precision in mathematical communication.