/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Very small amounts of time much ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 12

Very small amounts of time much less than a second have special names. Some of these names may be familiar to you, such as a millisecond, or \(0.001\) second. Have you heard of a nanosecond or a microsecond? A nanosecond is \(1 \times 10^{-9}\) second, and a microsecond is \(1 \times 10^{-6}\) second. How many nanoseconds are in a microsecond?

Short Answer

Expert verified
There are 1000 nanoseconds in a microsecond.

Step by step solution

01

Understanding Units

First, let's understand the units involved in the problem. A nanosecond is defined as \(1 \times 10^{-9}\) seconds, while a microsecond is \(1 \times 10^{-6}\) seconds. The task is to determine how many nanoseconds fit into a single microsecond.
02

Setting Up the Division

Since we are asked to find the number of nanoseconds in one microsecond, we should divide the value of one microsecond by the value of one nanosecond. This means we will compute the following: \( \frac{10^{-6}}{10^{-9}}\).
03

Simplifying the Expression

To divide powers of ten, we subtract the exponent of the divisor (nanoseconds) from the exponent of the dividend (microseconds). Therefore, \( 10^{-6} \div 10^{-9} = 10^{-6 - (-9)} = 10^{3} \).
04

Final Calculation

Once we have simplified the expression, we find that \(10^{3}\) nanoseconds are equivalent to one microsecond. This means there are 1000 nanoseconds in a microsecond.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nanosecond
A nanosecond is an incredibly tiny unit of time. It represents one billionth of a second, which is expressed as \(1 \times 10^{-9}\) seconds. Nanoseconds are often used in fields where timing needs to be extremely precise, such as in computer science and electronics. Understanding nanoseconds can help you appreciate the speed of modern technology, where every operation is often measured in these minuscule time units.

Consider computers and processors: they execute millions of instructions per second, and nanoseconds are necessary to measure these speeds accurately. This precision is what enables seamless digital communications and data processing.
  • 1 nanosecond = \(1 \times 10^{-9}\) seconds
  • Used in: computing, telecom industry, physics
  • Helps measure: processor speeds, data transfer rates
Microsecond
A microsecond is another unit of time, slightly larger than a nanosecond but still very brief. It is equal to one millionth of a second, or \(1 \times 10^{-6}\) seconds. Microseconds are used in applications that require high speed and precision but at scales where nanoseconds might be unnecessarily small.

For example, microseconds are important in transactions where data is verified almost instantaneously, such as in high-frequency trading on stock exchanges. They are also used in automotive and industrial applications for sensors and controls that respond in real-time.
  • 1 microsecond = \(1 \times 10^{-6}\) seconds
  • Used in: finance, industrial sensors, engineering
  • Important in: real-time processes, electronic communication
Exponentiation
Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent, meaning \(a\) is multiplied by itself \(b\) times. In the context of our time units, exponentiation helps express very large or very small numbers conveniently.

For instance, in the division of microseconds by nanoseconds, understanding exponents allows us to easily calculate \(\frac{10^{-6}}{10^{-9}}\). This is simplified by subtracting the exponents: \(-6 - (-9) = 3\), resulting in \(10^3\). With exponentiation, complex calculations become straightforward, enhancing accuracy and understanding in scientific computations.
  • Purpose: Express large/small quantities, simplify math operations
  • In our example: \(10^{-6} \div 10^{-9} = 10^3\)
  • Applications: sciences, engineering, technology

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Draw a "starting" line segment \(2 \mathrm{~cm}\) long on a sheet of paper. a. Draw a segment 3 times as long as the starting segment. How long is this segment? b. Draw a segment 3 times as long as the segment in 10 a. How long is this segment? c. Use the starting length and an exponent to write an expression that gives the length in centimeters of the next segment you would draw. (a) d. Use the starting length and an exponent to write an expression that gives the length in centimeters of the longest segment you could draw on a \(100 \mathrm{~m}\) soccer field.

In the course of a mammal's lifetime, its heart beats about 800 million times, regardless of the mammal's size or weight. (This excludes humans.) a. An elephant's heart beats approximately 25 times a minute. How many years would you expect an elephant to live? Use scientific notation to calculate your answer. (a) b. A pygny shrew's heart beats approximately 1150 times a minute. How many years would you expect a pygnry shrew to live? c. If this relationship were true for humans, how many years would you expect a human being with a heart rate of 60 bpm to live?

Use the properties of exponents to rewrite each expression. Use your calculator to check that your expression is equivalent to the original expression. [ \(\square\) See Calculator Note 6B to learn how to check equivalent expressions. 4] a. \(3 x^{2} \cdot 2 x^{4}\) b. \(5 x^{2} y^{3} \cdot 4 x^{4} y^{5}\) c. \(2 x^{2} \cdot 3 x^{3} y^{4}\) d. \(x^{3} \cdot 4 x^{4}\)

Use a recursive routine to find the first six terms of a sequence that starts with 100 and has a constant multiplier of \(-1.6\).

Because the number of molecules in a given amount of a compound is usually a very large number, scientists often work with a quantity called a mole. One mole is about \(6.02 \times 10^{23}\) molecules. a. A liter of water has about \(55.5\) moles of \(\mathrm{H}_{2} \mathrm{O}\). How many molecules is this? Write your answer in scientific notation. b. How many molecules are in \(6.02 \times 10^{23}\) moles of a compound? Write your answer in scientific notation.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.