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. \(84.3 mm\) b. \(2.41 m\) c. \(2.945 \times 10^{-5} cm\) d. \(14.45 km\) e. \(235300 mm\) f. \(0.9033 \mu m\)

Step-by-step solution

Identify the conversion factor

Since there are 10 millimeters in 1 centimeter, the conversion factor is 10.

Multiply the distance by the conversion factor

\(8.43 cm \times 10 = 84.3 mm\) b. Convert \(2.41 \times 10^{2} cm\) to meters

Identify the conversion factor

Since 1 meter = 100 centimeters, the conversion factor is 0.01.

Multiply the distance by the conversion factor

\((2.41 \times 10^{2}) cm \times 0.01 = 2.41 m\) c. Convert 294.5 nm to centimeters

Identify the conversion factor

Since 1 nm = \(10^{-9} m\) and 100 cm = 1 m, we can multiply both sides by 100 to get:1 nm = \(10^{-9} m \times 100 = 10^{-7} cm\)

Multiply the distance by the conversion factor

\(294.5 nm \times 10^{-7} = 2.945 \times 10^{-5} cm\) d. Convert \(1.445 \times 10^{4} m\) to kilometers

Identify the conversion factor

Since 1 km = 1000 meters, the conversion factor is 0.001.

Multiply the distance by the conversion factor

\((1.445 \times 10^{4} m) \times 0.001 = 14.45 km\) e. Convert 235.3 m to millimeters

Identify the conversion factor

Since 1 meter = 1000 millimeters, the conversion factor is 1000.

Multiply the distance by the conversion factor

\(235.3 m \times 1000 = 235300 mm\) f. Convert 903.3 nm to micrometers

Identify the conversion factor

Since 1 nm = \(10^{-9} m\) and 1 μm = \(10^{-6} m\), we can divide both by 1,000 to get:1 nm = \(\frac{10^{-9}}{10^{-6}} = 10^{-3} μm\)

Multiply the distance by the conversion factor

\(903.3 nm \times 10^{-3} = 0.9033 μm\)

Key concepts

Conversion Factors

Conversion factors are essential tools in the world of measurements, allowing us to seamlessly transition between different units of measurement. They are a cornerstone of mathematics and scientifically quantifying the world around us.

Think of conversion factors as mathematical bridges that help us relate two units, so we can easily convert one into the other. For instance, if we know that 1 inch is equivalent to 2.54 centimeters, we then have a conversion factor: 1 inch / 2.54 cm, or 2.54 cm / 1 inch, depending on the direction of the conversion.

It's like having the right key for a lock; once you have the conversion factor, you can unlock the value in a new unit. This is done by multiplying or dividing the quantity by the conversion factor, depending on whether you're converting to a larger or smaller unit respectively.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another using conversion factors. It’s a fundamental skill in both everyday life and scientific disciplines. To perform a unit conversion, you need to understand both the units you are converting from and to, and the relationship between them.

This process can be simplified into two steps:

1. First, identify the correct conversion factor to use.
2. Then, apply this factor to the quantity you are converting, ensuring that the units cancel appropriately so you are left with the desired unit.

For example, converting from centimeters to meters involves dividing by 100 because 1 meter equals 100 centimeters. The inverse is true when converting from meters to centimeters; you would multiply by 100.

Remember to consider significant figures and rounding as well, since precision can affect the result of your conversion, especially in scientific contexts.

Scientific Notation

Scientific notation is designed to handle the very large or very small numbers often encountered in science and engineering. By expressing numbers as a coefficient between 1 and 10, and multiplying it by ten raised to a power, scientific notation makes numbers easier to read, compare, and perform calculations with.

For example, the number 2,500 can be written in scientific notation as 2.5 x 10³. It simplifies complex calculations and provides a clear means of expressing precision through the number of significant digits in the coefficient.

In unit conversion, scientific notation is invaluable when dealing with measurements that span multiple orders of magnitude, like nanometers to kilometers. Understanding how to use scientific notation allows students to perform these conversions more effortlessly and accurately, avoiding the pitfalls of manually handling many zeroes.