Verified Answer :
- Convert the scientific notation \(2.54 \times 10^{-2} \, \text{m}\) to the decimal value \(0.0254 \, \text{m}\).
- Convert the result to centimeters by multiplying by \(100\), yielding \(2.54 \, \text{cm}\).
- Recognize that \(2.54 \, \text{cm}\) is the international standard for the length of exactly 1 inch.
- The distance represents the length of \(\boxed{1 \, \text{inch}}\).
Explanation
- Analyze the numerical value and convert to decimal.
The problem provides a distance between two points as \(2.54 \times 10^{-2} \, \text{m}\). To understand the significance of this value, we first convert the scientific notation into a standard decimal form. The expression \(2.54 \times 10^{-2}\) means that we shift the decimal point two places to the left, which gives:
\[ 2.54 \times 10^{-2} = 0.0254 \]
So, the distance is \(0.0254 \, \text{m}\).
- Convert the distance to centimeters.
In the metric system, we know that \(1 \, \text{m} = 100 \, \text{cm}\). To convert the distance from meters to centimeters, we multiply the value in meters by 100:
\[ 0.0254 \, \text{m} \times 100 \, \frac{\text{cm}}{\text{m}} = 2.54 \, \text{cm} \]
Thus, the distance is \(2.54 \, \text{cm}\).
- Identify the physical significance of the distance.
Historically, the inch is a unit of length in the imperial and US customary systems of measurement. Since 1959, the international inch has been defined as exactly \(25.4\) millimeters. Since \(25.4 \, \text{mm} = 2.54 \, \text{cm}\), the distance \(2.54 \, \text{cm}\) is exactly equivalent to 1 inch. Therefore, the given distance of \(2.54 \times 10^{-2} \, \text{m}\) corresponds to the length of 1 inch.
Examples
Understanding unit conversions is fundamental in various fields, such as engineering and international trade. For example, if a manufacturer in the United States produces screws measured in inches, they must accurately convert these dimensions into centimeters or meters to ensure compatibility with parts manufactured in countries that use the metric system. Recognizing that \(2.54 \, \text{cm} = 1 \, \text{inch}\) allows for precise communication and manufacturing across global borders.