Question

Set up the problem using dimensional analysis and make the conversion as indicated. 8 in = ______ \(\mathrm{cm}\)

Short answer

8 inches is equal to 20.32 centimeters.

Step-by-step solution

Identify the Conversion Factors

To convert inches to centimeters, you need to know the relationship between these two units. The conversion factor is 1 inch = 2.54 centimeters.

Set Up the Conversion Equation

Write the given measurement in inches and multiply it by the conversion factor to convert it to centimeters. The equation will be:\[8 \, \text{inches} \times \frac{2.54 \, \text{cm}}{1 \, \text{inch}}\]

Perform the Multiplication

Carry out the multiplication of the values:\[8 \, \text{inches} \times 2.54 \, \text{cm} = 20.32 \, \text{cm}\]

Cancel Out the Units

In this step, realize that the unit 'inches' in the numerator and the denominator cancels out, leaving you with the result in centimeters. Thus, the final measurement is 20.32 cm.

Key concepts

Unit Conversion

Unit conversion is a mathematical technique used to convert a quantity expressed in one unit to another. This is essential in ensuring that measurements are consistent and comparable across different systems or contexts. To perform a unit conversion, you need to multiply the original measurement by a conversion factor. This factor is a fraction that represents the relationship between the units involved.

For example, when converting inches to centimeters, as in the exercise, the conversion factor is 1 inch = 2.54 centimeters. This means that for every inch, there are 2.54 centimeters. The factor \(\frac{2.54 \, \text{cm}}{1 \, \text{inch}}\) is used to change the unit from inches to centimeters.

Key points:

• Conversion keeps proportional relationships between units.
• The unit to cancel is placed in the opposite part of the fraction (numerator or denominator) as the original measurement.
• After canceling out the original unit, the measurement is expressed in the new desired unit.

Measurement

Measurement is the process of obtaining the size, length, or amount of something typically using established units. It is fundamental to science, engineering, construction, and many other fields. Measurements must be precise and accurate to avoid errors in data interpretation or application.

When measuring physical quantities, like length, it is important to choose the correct unit. For instance, the imperial system uses inches, whereas the metric system uses centimeters. Knowing how to adjust between these systems forms the basis for understanding and performing unit conversions effectively. In this context, the original problem requires the understanding of a basic yet crucial measurement conversion from inches to centimeters.

Tips for accurate measurement:

• Always use calibrated tools for precision.
• Ensure readings are done at eye level to avoid parallax errors.
• Document readings promptly and clearly to maintain accuracy.

Inches to Centimeters

Converting inches to centimeters is one of the most common unit conversions due to the widespread use of both units in different parts of the world. The United States primarily uses inches, while most other countries operate within the metric system using centimeters.

The conversion formula is straightforward: multiply the number of inches by 2.54 to obtain the length in centimeters. So when you have a measurement of 8 inches, as shown in the exercise, you multiply 8 by 2.54 to get 20.32 centimeters.

Reasons to convert:

• Ensures compatibility with international standards and reports.
• Facilitates communication of measurements in scientific and technical contexts.
• Helps when working with international teams or datasets that may use different units.