Problem 80 - Beginning and Intermediate Algebra

Question

A nanometer is \(0.000000001 \mathrm{m}\) Scientists have developed optical nanowires to transmit light waves short distances. A nanowire with a diameter of 360 nanometers has been used in experiments on the transmission of light (Source: The New York Times, January 29, 2004). Find the diameter of such a wire, in meters.

Step-by-Step Solution

Answer

The diameter of the nanowire in meters is \(3.6 \times 10^{-7} \mathrm{m}\).

Step 1: Write the conversion factor

The conversion factor for nanometers to meters is 1 nanometer = \(0.000000001 \mathrm{m}\).

Step 2: Write the given diameter in nanometers

The diameter of the nanowire is given as 360 nanometers.

Step 3: Convert the diameter from nanometers to meters

To convert the diameter from nanometers to meters, we will multiply the given diameter in nanometers by the conversion factor: Diameter in meters = Diameter in nanometers × Conversion factor Diameter in meters = 360 nanometers × \(0.000000001 \mathrm{m}\)/nanometer

Step 4: Calculate the diameter in meters

Diameter in meters = 360 × \(0.000000001 \mathrm{m}\) Diameter in meters = \(360 \times 10^{-9} \mathrm{m}\) Diameter in meters = \(3.6 \times 10^{-7} \mathrm{m}\) So, the diameter of the nanowire in meters is \(3.6 \times 10^{-7} \mathrm{m}\).

Key Concepts

Nanometers to Meters

When dealing with very small lengths, such as those measured in nanometers, it becomes important to know how to convert these measurements to a more commonly used unit like meters. A nanometer is a millionth of a millimeter, or equivalently, one-billionth of a meter. In scientific notation, this is expressed as \(1 \times 10^{-9}\) meters, showing how incredibly tiny a nanometer is.

To convert nanometers into meters, you simply multiply the number of nanometers by this conversion factor (\(1 \times 10^{-9}\) meters per nanometer). For example, if you have an object that measures 360 nanometers, you convert it by calculating:\[ \text{Diameter in meters} = 360 \times 10^{-9} \text{ meters} \]This ultimately gives you \(3.6 \times 10^{-7}\) meters. This conversion is a routine task in scientific calculations, helping scientists and engineers communicate their findings using standardized units.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It uses powers of ten to simplify these expressions. A number in scientific notation looks like \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer. This method is highly efficient for calculations and is used frequently in scientific and engineering contexts.

For example, instead of writing 0.000000001 in standard form, scientific notation lets us write it as \(1 \times 10^{-9}\). This not only makes the number easier to read and write, but it also simplifies mathematical operations like multiplication and division. In the context of the nanowire, the diameter converts to \(3.6 \times 10^{-7}\) meters using scientific notation, highlighting its practicality in everyday scientific use.

Optical Nanowires

Optical nanowires are fascinating components used in the field of nanotechnology and optics. They act as ultra-thin fibers that guide light with high precision over very short distances. These structures are crucial for advanced technological applications, such as communication systems, biosensing, and computational devices.

The precision with which they can transmit light makes them vital for manipulating optical signals in a controlled environment. For the given problem, understanding that the diameter of such a nanowire is 360 nanometers gives insight into the high degree of control needed at a nanoscale when handling optical technologies. These nanowires serve as a perfect example of how advancements in nanotechnology can revolutionize the way we use and transmit information at the smallest scales possible.