Question

Perform each of the following conversions. a. 8.43 \(\mathrm{cm}\) to millimeters b. \(2.41 \times 10^{2} \mathrm{cm}\) to meters c. 294.5 \(\mathrm{nm}\) to centimeters d. \(1.445 \times 10^{4} \mathrm{m}\) to kilometers e. 235.3 \(\mathrm{m}\) to millimeters f. 903.3 \(\mathrm{nm}\) to micrometers

Short answer

a. \(8.43\, \mathrm{cm} = 84.3\, \mathrm{mm}\) b. \(2.41 \times 10^{2}\, \mathrm{cm} = 2.41\, \mathrm{m}\) c. \(294.5\, \mathrm{nm} = 2.945\times 10^{-7}\, \mathrm{cm}\) d. \(1.445 \times 10^{4}\, \mathrm{m} = 14.45\, \mathrm{km}\) e. \(235.3\, \mathrm{m} = 235300\, \mathrm{mm}\) f. \(903.3\, \mathrm{nm} = 0.9033\, \mathrm{\mu m}\)

Step-by-step solution

Identify the conversion factor between centimeters and millimeters

There are 10 millimeters in 1 centimeter.

Perform the conversion

Multiply 8.43 cm by the conversion factor (10 millimeters/1 centimeter): \[8.43 \, \mathrm{cm} \times \frac{10 \, \mathrm{mm}}{1 \, \mathrm{cm}} = 84.3 \, \mathrm{mm}\] #b. Convert \(2.41 \times 10^{2} \, \mathrm{cm}\) to meters#

Identify the conversion factor between centimeters and meters

There are 100 centimeters in 1 meter.

Perform the conversion

Divide \(2.41 \times 10^{2} \, \mathrm{cm}\) by the conversion factor (100 cm/1 meter): \[\frac{2.41 \times 10^{2}\, \mathrm{cm}}{100\, \mathrm{cm/m}} = 2.41\, \mathrm{m}\] #c.Convert 294.5 nm to centimeters#

Identify the conversion factor between nanometers and centimeters

There are \(1 \times 10^9\) nanometers in 1 meter and 100 centimeters in 1 meter.

Perform the conversion

Divide 294.5 nm by the conversion factor (1e9 nm/1 m) and multiply by 100 cm/1 m: \[\frac{294.5\, \mathrm{nm}}{1\times 10^9\, \mathrm{nm/m}} \times \frac{100\, \mathrm{cm}}{1\, \mathrm{m}} = 2.945\times 10^{-7}\, \mathrm{cm}\] #d. Convert \(1.445 \times 10^{4}\, \mathrm{m}\) to kilometers#

Identify the conversion factor between meters and kilometers

There are 1000 meters in 1 kilometer.

Perform the conversion

Divide \(1.445 \times 10^{4}\, \mathrm{m}\) by the conversion factor (1000 m/1 km): \[\frac{1.445\times 10^{4}\,\mathrm{m}}{1000\, \mathrm{m/km}} = 14.45\, \mathrm{km}\] #e. Convert 235.3 m to millimeters#

Identify the conversion factor between meters and millimeters

There are 1000 millimeters in 1 meter.

Perform the conversion

Multiply 235.3 m by the conversion factor (1000 mm/1 m): \[235.3\, \mathrm{m} \times \frac{1000\, \mathrm{mm}}{1 \, \mathrm{m}} = 235300\, \mathrm{mm}\] #f. Convert 903.3 nm to micrometers#

Identify the conversion factor between nanometers and micrometers

There are 1000 nanometers in 1 micrometer.

Perform the conversion

Divide 903.3 nm by the conversion factor (1000 nm/1 μm): \[\frac{903.3\, \mathrm{nm}}{1000\, \mathrm{nm/\mu m}} = 0.9033\, \mathrm{\mu m}\]

Key concepts

Metric System

The metric system is a decimal-based system of measurement used widely around the world. It simplifies calculations by using the power of ten. Unlike other systems, the metric system is straightforward due to its standardized units and prefixes.
For example:

• Kilo- means 1000 times the base unit.
• Centi- means 1/100th of the base unit.
• Milli- means 1/1000th of the base unit.

This method of naming ensures that converting between units is seamless and easy to understand.
In daily life and science fields, the metric system standardizes measurements in length, volume, mass, and other quantities. This system's ease of use is beneficial for educational purposes, trade, and global scientific research.

Conversion Factors

Conversion factors play a crucial role in making various unit conversions effortless. They provide the necessary multiplier or divisor to switch from one unit to another, ensuring accuracy and consistency in measurements.
Understanding these factors is essential for converting measurements in the metric system. Some key examples include:

• 1 centimeter is equal to 10 millimeters.
• 1 meter is equal to 100 centimeters.
• 1 kilometer is equal to 1000 meters.
• 1 millimeter is equal to 1000 micrometers.
• 1 nanometer is equal to 0.000001 millimeters.

To convert, always multiply or divide by these conversion factors, depending on whether you need larger or smaller units. For instance, when converting from a larger unit to a smaller one, you multiply, and vice versa. This systematic approach ensures that all conversion tasks are manageable and clear.

Length Measurement

Length measurement in the metric system involves using units such as millimeters, centimeters, meters, and kilometers. Each unit serves specific needs, from measuring tiny objects to vast distances.

• Millimeters are used for very small lengths, like the thickness of a credit card.
• Centimeters are useful for everyday objects, like the width of a smartphone.
• Meters are practical for room dimensions and small distances.
• Kilometers are great for measuring larger distances, like the distance between two cities.

Nanometers, also within the metric system, help in measuring atomic scale lengths, often seen in scientific research.
Understanding these units' applications ensures precise communication and measurement across various fields and disciplines. Knowing how to convert between these lengths using the metric system is vital for accuracy and uniformity.