Problem 76 - Chemistry Principles and Practice

Question

A standard sheet of paper in the United States is \(8.5 \times 11\) inches. Express the area of this sheet of paper in square centimeters.

Step-by-Step Solution

Answer

The area of the sheet is approximately 603.23 cm².

Step 1: Understand the Conversion Factor

To convert from square inches to square centimeters, we need to know the conversion factor: 1 inch is equal to 2.54 centimeters. Therefore, to convert an area, we square this conversion factor.

Step 2: Calculate the Conversion Factor for Area

Calculate how many square centimeters are in one square inch by squaring the conversion factor from inches to centimeters. The conversion factor is:\[(2.54 \, \text{cm})^2 = 6.4516 \, \text{cm}^2\]

Step 3: Find the Area in Square Inches

Calculate the area of the sheet in square inches by multiplying the length and width:\[8.5 \, \text{inches} \times 11 \, \text{inches} = 93.5 \, \text{square inches}\]

Step 4: Convert to Square Centimeters

Multiply the area in square inches by the conversion factor to get the area in square centimeters:\[93.5 \, \text{square inches} \times 6.4516 \, \text{cm}^2/\text{square inch} = 603.2256 \, \text{cm}^2\]

Step 5: Round the Result

Round the calculated area to a reasonable degree of precision. Thus, the area of the paper in square centimeters is approximately 603.23 cm².

Key Concepts

Area Calculation

Calculating the area of a surface involves determining the amount of space it occupies within a boundary. In the case of a rectangular object, like a standard sheet of paper, the area is calculated by multiplying its length by its width. For instance, a paper that measures 8.5 inches in width and 11 inches in height will have an area of

• Width = 8.5 inches
• Height = 11 inches
• Area = Width × Height = 8.5 × 11 = 93.5 square inches

This process uses the basic formula for the area of a rectangle: Area = length × width. Understanding this simple multiplication is key to solving more complex problems where the surface shape or units might differ.
Area calculation ensures you can determine the coverage or size for various practical applications, from fitting papers into frames to planning architectural layouts.

Measurement Conversion

Unit conversion methods are vital when comparing measurements in different units. In this exercise, we are converting from square inches to square centimeters. Since 1 inch is equal to 2.54 centimeters, for area, we must square the linear conversion factor.

• 1 inch = 2.54 cm
• 1 square inch = (2.54 cm) × (2.54 cm) = 6.4516 cm²

This factor of 6.4516 cm² is crucial to accurately convert square inch measurements into square centimeters, ensuring precision in calculations.
Conversion becomes important in a diverse range of scenarios, such as engineering designs, medical tasks, and even when traveling between countries using different measurement systems. It guarantees uniformity and consistency in communicating sizes and dimensions.

Metric System

The metric system is a decimal-based system of measurement used globally, making conversions intuitive and straightforward. It uses units like meters, liters, and grams, which are all based on powers of ten. In conversions, this simplifies calculations compared to non-metric systems.

• Easy calculations with powers of ten
• Global standardization improves understanding across different regions

For example, the conversion from centimeters to meters is straightforward: 1 meter equals 100 centimeters. This decimal characteristic makes the metric system immensely beneficial in scientific, medical, and educational fields.
Understanding the metric system aids in learning measurement conversions, such as from inches to centimeters, allowing for clear comparison and understanding of measurements worldwide. Mastery of this system opens up effective communication and interoperability among international communities.