Problem 64 - Technical Mathematics with Calculus

Question

A generator has an output of \(5.34 \times 10^{6}\) millivolts. What is the output in kilovolts?

Step-by-Step Solution

Answer

The output in kilovolts is 5.34 kV.

Step 1: Convert millivolts to volts

Since there are 1,000 millivolts in one volt, divide the number of millivolts by 1,000 to convert to volts. So the calculation is: \[\begin{equation} 5.34 \times 10^{6} \, \text{millivolts} \times \frac{1 \, \text{volt}}{1000 \, \text{millivolts}} \end{equation}\]

Step 2: Convert volts to kilovolts

As there are 1,000 volts in one kilovolt, divide the voltage by 1,000 to convert to kilovolts. So the calculation is: \[\begin{equation} \frac{5.34 \times 10^{6} \, \text{millivolts}}{1000} \times \frac{1 \, \text{kilovolt}}{1000 \, \text{volts}} \end{equation}\]

Step 3: Calculate the output in kilovolts

By performing the multiplication and division, we find the output in kilovolts. The final calculation is: \[\begin{equation} 5.34 \times \frac{10^{6}}{1000} \times \frac{1}{1000} \, \text{kilovolts} \end{equation}\]

Key Concepts

Electrical Units Conversion

Understanding electrical units conversion is crucial for anyone working with electrical systems or studying physics and engineering. The basic unit of electrical potential or voltage is the volt (V). When dealing with a wide range of voltages, from very small to very large, it's efficient to use prefixes, making the numbers easier to work with. For instance, a millivolt (mV) is one thousandth of a volt, whereas a kilovolt (kV) is one thousand volts. Converting between these different scales helps make sense of electrical measurements, such as the output of a generator.

To convert millivolts to kilovolts in our exercise, we need to first convert millivolts to volts and then volts to kilovolts. Since the factor between millivolts and volts is 1,000, you divide the number of millivolts by 1,000. To then convert volts to kilovolts, you divide again by 1,000, as there are another 1,000 volts in a kilovolt. Through these steps, we make the scale of the measurement more meaningful for practical applications, like comparing outputs of different generators.

Scientific Notation

Scientific notation allows us to express either very large or very small numbers succinctly. In this system, numbers are written as the product of two factors: a coefficient that is greater or equal to 1 and less than 10, and a power of 10. For example, the number 53,400,000 can be written as \(5.34 \times 10^{7}\). This form of notation is particularly useful in scientific calculations, where measurements can span many orders of magnitude.

In the generator exercise, the output is given as \(5.34 \times 10^{6}\) millivolts in scientific notation, which is more concise than writing out all the zeros. When converting to kilovolts, the exponentiation rules simplify the process: dividing the coefficient by 1,000 twice effectively reduces the power of ten from \(10^{6}\) to \(10^{3}\), or simplifies the expression to \(5.34 \times 10^{3}\) kilovolts.

Metric Prefixes

Metric prefixes represent quantities on a standardized scale and are based on powers of ten. These prefixes attach to base units to indicate multiplication or division by a specific factor. For example, 'kilo-' signifies a factor of one thousand, and 'milli-' signifies a factor of one thousandth. This system is incredibly efficient for describing units of measure such as length, mass, and, as relevant to our exercise, voltage.

Through the use of metric prefixes, we can easily scale units up or down, making it easier to grasp the magnitude of a measurement and perform conversions. In the context of the generator output problem, understanding that 'milli-' means dividing by one thousand, and 'kilo-' means multiplying by one thousand, is essential for accurate unit conversion.