Question

A human hair has a thickness of about \(70 \mu \mathrm{m}\). What is this thickness in (a) meters and (b) kilometers?

Short answer

The thickness is \(7.0 \times 10^{-5}\) meters and \(7.0 \times 10^{-8}\) kilometers.

Step-by-step solution

Understand the Given Measurement

The thickness of the hair is given as \(70 \mu \mathrm{m}\), which represents 70 micrometers. One micrometer is equal to \(10^{-6}\) meters.

Convert Micrometers to Meters

To convert \(70 \mu \mathrm{m}\) to meters, multiply by \(10^{-6}\):\[70 \mu \mathrm{m} = 70 \times 10^{-6} \text{ meters}\]. This equals \(7.0 \times 10^{-5}\) meters.

Convert Meters to Kilometers

To convert meters to kilometers, use the conversion factor \(1 \text{ kilometer} = 1000 \text{ meters}\). Thus, divide \(7.0 \times 10^{-5}\) meters by 1000: \[7.0 \times 10^{-5} \div 1000 = 7.0 \times 10^{-8} \text{ kilometers}\].

Key concepts

Micrometers to Meters

When converting measurements, it is key to know the relationships between units. Micrometers (\(\mu m\)) are vastly smaller than meters. It's essential to note that "micro" denotes a factor of a millionth, or\(10^{-6}\). To convert from micrometers to meters, multiply the number of micrometers by \(10^{-6}\). For instance, a hair measuring \(70\,\mu m\) requires multiplying 70 by \(10^{-6}\). - This results in\(70\, \times \, 10^{-6} = 7.0 \times 10^{-5}\) meters. Always ensure your result is in scientific notation for simplicity and precision, especially with tiny measurements like these. Remember that this conversion helps in translating microscopic dimensions to a more standard unit, aiding in better understanding and comparison.

Meters to Kilometers

Converting meters to kilometers involves a simpler relationship since both are standard units of length. One kilometer consists of 1000 meters. Conversion, therefore, involves division. For example, if something measures \(7.0 \times 10^{-5}\) meters, convert it to kilometers by dividing by 1000. This operation looks like this:\[7.0 \times 10^{-5} \div 1000\]- The result is\(7.0 \times 10^{-8}\) kilometers. Breaking measurements down through division simplifies understanding, especially with such small quantities. It also emphasizes the nature of kilometers being thousands of a meter, helping relate various measurement scales effectively.

Measurement Conversion

Measurement conversion is an essential skill in science and everyday situations. Understanding the relationship between different units allows for accurate descriptions and comparisons.- **Fundamentals of Conversion**: - Through multiplication or division by conversion factors, measurements convert from one unit to another. - Conversion factors are typically standardized, ensuring consistent results. - **Practical Example**: Converting \(70 \mu m\) to meters and then to kilometers emphasizes using multiple conversions in one problem, showcasing cumulative understanding of unit scales. Effective conversion is crucial for problem-solving in scientific and engineering contexts. It aids significantly in communication and representation of data across different unit systems, providing a common framework for understanding distances, lengths, and other quantities.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers succinctly. It uses a base number and an exponent on 10 to represent these quantities effectively.In the hair thickness example:- \(70 \mu m\) is converted to \(7.0 \times 10^{-5}\) meters.- Similarly, this translates to \(7.0 \times 10^{-8}\) kilometers. Scientific notation helps manage unwieldy numbers, making them easier to read, write, and compare. It is especially used in science and engineering, where measurements often involve extreme values. Getting comfortable with shifting between standard numbers and scientific notation streamlines calculations and aids in maintaining precision and clarity across computations.