Chapter 1, Problem 122QP

Rank the following measurements in order from smallest to largest: $1.0\times {10}^{-4}\text{m},2.0\times {10}^{-5}\text{m},3.0\times {10}^{-6}\text{km},4.0\times {10}^2\text{mm,}0.0\text{m},1.0\text{m}.$

Interpretation Introduction

Interpretation:

The given measurements are to be arranged in the increasing order.

Concept Introduction:

Unit conversion is the method used to convert one unit of quantity into another. There are conversions factors of each of the units are provided through which the conversion of the unit is taken place.

Answer to Problem 122QP

Solution:

$1.0\times {10}^{-4}\text{m}<2.0\times {10}^{-5}\text{m}<3.0\times {10}^{-3}\text{m}<4.0\times {10}^{-1}\text{m}<\text{0}\text{.0 m}<1.0\text{ m}$

Explanation of Solution

Given Information: $1.0\times {10}^{-4}\text{m}$ , $2.0\times {10}^{-5}\text{m}$ , $3.0\times {10}^{-6}\text{ km}$ . $4.0\times {10}^{-1}\text{mm}$ , $\text{0}\text{.0 m}$ , $1.0\text{ m}$ .

The units of all the measurements should be equal to be able to compare and arrange them in the increasing order.

The conversion of $3.0\times {10}^{-6}\text{ km}$ into $\text{m}$ is:

The relation between kilometer $\left(\text{km}\right)$ and meter $\left(\text{m}\right)$ is shown below.

$1\text{ km}=1000\text{ m}$

The conversion factors used to convert $\text{km}$ into $\text{m}$ are shown below.

$\frac{1\text{ km}}{1000\text{ m}}\text{or}\frac{1000\text{ m}}{1\text{ km}}$

$3.0\times {10}^{-6}\text{ km}$ is converted into $\text{m}$ using a conversion factor, as shown below.

$\begin{array}{c}3.0\times {10}^{-6}\text{ km}=3.0\times {10}^{-6}\text{ km}\times \frac{1000\text{ m}}{1\text{ km}}\\ =3.0\times {10}^{-3}\text{m}\end{array}$

Therefore, $3.0\times {10}^{-6}\text{ km}$ is equal to $3.0\times {10}^{-3}\text{m}$ .

The conversion of $4.0\times {10}^2\text{ mm}$ to $\text{m}$ is:

The relation between millimeter $\left(\text{mm}\right)$ and meter $\left(\text{m}\right)$ is shown below.

$1\text{ mm}={10}^{-3}\text{ m}$

The conversion factors used to convert $\text{mm}$ into $\text{m}$ is shown below.

$\frac{1\text{ mm}}{{10}^{-3}\text{ m}}\text{or}\frac{{10}^{-3}\text{ m}}{1\text{ mm}}$

$4.0\times {10}^2\text{ mm}$ is converted into $\text{m}$ using a conversion factor, as shown below.

$\begin{array}{c}4.0\times {10}^2\text{ mm}=4.0\times {10}^2\text{ mm}\times \frac{{10}^{-3}\text{ m}}{1\text{ mm}}\\ =4.0\times {10}^{-1}\text{m}\end{array}$

Therefore, $4.0\times {10}^2\text{ mm}$ is equal to $4.0\times {10}^{-1}\text{m}$ .

The given measurements are $1.0\times {10}^{-4}\text{m}$ , $2.0\times {10}^{-5}\text{m}$ , $3.0\times {10}^{-3}\text{m}$ , $4.0\times {10}^{-1}\text{m}$ , $\text{0}\text{.0 m}$ , and $1.0\text{ m}$ .

The measurements are arranged in the increasing order as follows:

$1.0\times {10}^{-4}\text{m}<2.0\times {10}^{-5}\text{m}<3.0\times {10}^{-3}\text{m}<4.0\times {10}^{-1}\text{m}<\text{0}\text{.0 m}<1.0\text{ m}$

Therefore, this is the increasing order of the given measurements.

Conclusion

The measurements are arranged in the increasing order is as follows:

$1.0\times {10}^{-4}\text{m}<2.0\times {10}^{-5}\text{m}<3.0\times {10}^{-3}\text{m}<4.0\times {10}^{-1}\text{m}<\text{0}\text{.0 m}<1.0\text{ m}$